<%BANNER%>

Input/Output Control of Asynchronous Machines with Races

Permanent Link: http://ufdc.ufl.edu/UFE0021263/00001

Material Information

Title: Input/Output Control of Asynchronous Machines with Races
Physical Description: 1 online resource (81 p.)
Language: english
Creator: Peng, Jun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: asynchronous, control, feedback, input, machine, output
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The occurrence of races causes unpredictable and undesirable behavior in asynchronous sequential machines. In the present work, traditional feedback control techniques are used to control a race-afflicted machine, so as to turn it into a deterministic machine that matches a desired model. Instead of replacing or redesigning the whole machine, I add an output feedback controller to the original defective machine, and the controller eliminates the negative effects of the critical races. The present work focuses on asynchronous sequential machines in which the state of the machine is not provided as an output. The results include the necessary and sufficient conditions for the existence of controllers that eliminate the effects of a critical race, as well as algorithms for their design. The necessary and sufficient conditions for the existence of controllers are presented in terms of certain matrix inequalities.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jun Peng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hammer, Jacob.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021263:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021263/00001

Material Information

Title: Input/Output Control of Asynchronous Machines with Races
Physical Description: 1 online resource (81 p.)
Language: english
Creator: Peng, Jun
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: asynchronous, control, feedback, input, machine, output
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The occurrence of races causes unpredictable and undesirable behavior in asynchronous sequential machines. In the present work, traditional feedback control techniques are used to control a race-afflicted machine, so as to turn it into a deterministic machine that matches a desired model. Instead of replacing or redesigning the whole machine, I add an output feedback controller to the original defective machine, and the controller eliminates the negative effects of the critical races. The present work focuses on asynchronous sequential machines in which the state of the machine is not provided as an output. The results include the necessary and sufficient conditions for the existence of controllers that eliminate the effects of a critical race, as well as algorithms for their design. The necessary and sufficient conditions for the existence of controllers are presented in terms of certain matrix inequalities.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jun Peng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hammer, Jacob.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021263:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101206_AAAACM INGEST_TIME 2010-12-06T13:12:53Z PACKAGE UFE0021263_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 1638 DFID F20101206_AABCVP ORIGIN DEPOSITOR PATH peng_j_Page_54.txt GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
b1945b6c04b29379fb28ff7fb065f370
SHA-1
e3691ea54ca925b876e3a75f683bd6f66297949b
78922 F20101206_AABCGX peng_j_Page_44.jp2
f2eebb6c4b4e44026f92c5c1b5dca960
ec8c258f29d0dcacdb6787763a0ed2d63e0031cb
1053954 F20101206_AABCQS peng_j_Page_65.tif
44e0833c64fc7f01cf44ae67c1defe09
301cbac067e5cc42827f3bf95cba9bab37d591cf
353 F20101206_AABCVQ peng_j_Page_55.txt
566cb1ba2c8cba97b3ea32188fb56a78
f14660163e21de6bceb3b08e2fdf777e927c1b16
67397 F20101206_AABCLV peng_j_Page_67.jpg
fef208e3331733122c7200f9fe40d52a
087c90b0e0dbcef610258c848ab97009d696e3bd
74318 F20101206_AABCGY peng_j_Page_13.jp2
491f4dc1f799e7966b0b2f42314cc576
da7dca0db665f6d447455cad02883858ecde7009
F20101206_AABCQT peng_j_Page_66.tif
02652028942c28279ed053da0d86b33d
a98d36258a5e369285854e7d1a26293e35ec24ec
2141 F20101206_AABCVR peng_j_Page_56.txt
19893628bf5db9d675b3ee78e5f4490c
3266ef754328fdf7aef65b3a7ba47496c4915941
40157 F20101206_AABCLW peng_j_Page_68.jpg
03deea7c9b2da2cda117d4275711c498
c9021ea4632da4e6280ff66b98a307e066808ba7
1900 F20101206_AABCGZ peng_j_Page_14.txt
59566f6a521cbd08127a1d2e0aad85f2
7d049159742e742ed0105e487a8a333a0781345f
F20101206_AABCQU peng_j_Page_67.tif
e620331083dce695e9e1510dac5ecd6e
9bf780b9ed4960d46649de6ada5b44cf49a1b2d9
1824 F20101206_AABCVS peng_j_Page_57.txt
bc8c8fcbec7b4fb00e97f1826c390b25
a54bc25ea8ab76e1f247331fb24bfc2c830b1355
51364 F20101206_AABCLX peng_j_Page_69.jpg
cc457728977d86fc5664be8bc1195452
3e4ff2df8bb71401f590769d9e574c98c604ba39
F20101206_AABCQV peng_j_Page_68.tif
8333cf264c1210e39480e75709848d2c
4696e7c7f23185a45f2d65241d3295dbd75d7be0
1599 F20101206_AABCVT peng_j_Page_58.txt
0327c1f039d2eecc244242e8a2844f78
a715f0b2d999d19a660d9529a9ad1dcde0b992ca
7464 F20101206_AABCJA peng_j_Page_41thm.jpg
0c53ccbd103fc2c0ecb4f6d1c73ab62c
566983dc4626fa1bbad4212d3e5e9311771f0ba8
66082 F20101206_AABCLY peng_j_Page_70.jpg
a6dc62d85ab6a3951f15d4404892d169
46ed563d0a7938bdb19146a2a96c209c3997f67a
F20101206_AABCQW peng_j_Page_70.tif
19defafb260e4ba1fe4c64b2e374d2c8
f2926f2fe012791a9d628c6611e113f8397f9b23
1283 F20101206_AABCVU peng_j_Page_59.txt
2f11f0b38ee98b4f8ac75633fd9dae0c
c0c1bb4233f8bc66104a48adb5627953ccaaf063
9149 F20101206_AABCJB peng_j_Page_78thm.jpg
a720f48b57ac63aa4f79d949463887fe
c5168abc650e9ff099554530a22af92314d41bc8
65658 F20101206_AABCLZ peng_j_Page_71.jpg
7b86309fb05b43321dd6ea44a82f1564
676be0232add294190acaec67fe5d77f12ac26be
F20101206_AABCQX peng_j_Page_71.tif
eb32a5d8d489a1ee518bc3a8618ade6f
c2a75afe386f9184a600dbad02d620fb654a0492
2053 F20101206_AABCVV peng_j_Page_60.txt
1a0075027acf9b357104fc4fc0c21268
8dc07759d5965db54acb33559353554a01ad9db3
78448 F20101206_AABCJC peng_j_Page_49.jp2
b875bba49063db3762fdef9d75580c46
d724a5a6c5e309fdb2b1f31b47046a3fb4564e3b
F20101206_AABCQY peng_j_Page_72.tif
78cbbe0eb31075b0794b8f822fca26ec
b98478b9cc039469ef4a3c06af7ef3da0607fbd6
1694 F20101206_AABCVW peng_j_Page_61.txt
4d8dd60f1790e7985bf8c96b41c89df2
904da251a6eefd7062fab175b2f01f545fe9e20b
101030 F20101206_AABCOA peng_j_Page_62.jp2
90336f54f6907b53fbb6ffc85676b1c8
25dc7c6e11508862096945af8fea390dfc008060
F20101206_AABCQZ peng_j_Page_73.tif
a2f579a42ca7be0a11af31762cdfd129
580664a5c79394aff42db0d4ac6758965da751fd
6404 F20101206_AABCJD peng_j_Page_70thm.jpg
5c7ecb9724fdc90b11e981b7da262be7
cab168c21a485f4f45c7ec43bc282d586b7f2733
1965 F20101206_AABCVX peng_j_Page_62.txt
745e7b230ae8f3e0a26f7359361fd52c
211a28b2dba9499e13db9fa21f0e2ce3b30ceede
86829 F20101206_AABCOB peng_j_Page_63.jp2
fada0b7bdab8c0674caae8f52da6d7c5
cc0e510089175bf5264ad73dbb4b4c4f2faee116
35111 F20101206_AABCJE peng_j_Page_74.QC.jpg
47684cdeb6076b6ccf407e25dd2bd9a6
7c5544ea1b73fcb8e2ea45af236493be6ccf7045
1774 F20101206_AABCVY peng_j_Page_63.txt
bff576162e07a6d36d14b0a1b8887ccc
fc519f72a1328a9ffd045c15c4d4c9d1d4807452
78593 F20101206_AABCOC peng_j_Page_64.jp2
16abb6d89c94097dbbfbd36d9df9117b
fb987a0884408b21d71dd27b262e42fd37ce9c63
890 F20101206_AABCJF peng_j_Page_04.txt
84dab869e02f1825e5ad173174ae625f
fe5484b3133a1175a10f220245b14faf58eda47f
1627 F20101206_AABCVZ peng_j_Page_64.txt
d9096ae33c0d48323b89aba11d516116
373d4c0fdf3d00ccf0727cfd13a6764fc60739ec
8300 F20101206_AABCTA peng_j_Page_55.pro
ef1b67eb4d047c2b847fdfa4558b044e
8de96283c14637699d0f67bd8574acf9a1c6cd09
85536 F20101206_AABCOD peng_j_Page_65.jp2
77ca47642321139223cae4042b77facc
9863225c8221ed5e7fdec508632e969e0d62aa74
101581 F20101206_AABCJG peng_j_Page_27.jp2
d94d8f259a8e451255540350c6d50766
90a4dfc5d04d71d191ec471a26c3b9debf1c96eb
51528 F20101206_AABCTB peng_j_Page_56.pro
44d66d4616bea2ce24c1aca3230ca060
4ec244b9451c6058179a45a4b2dfc897d37ed9ba
98053 F20101206_AABCOE peng_j_Page_66.jp2
7e528be174090ed4aa23da48d14cff91
05b5352c243190babf4b71686efed8692a9a8644
5192 F20101206_AABCJH peng_j_Page_40thm.jpg
5a9c89393c31f11dd23c5d64fc45e384
eef4879418c8d61d0a62ef4b9ac2e9a6e117fa4f
6918 F20101206_AABDAA peng_j_Page_53thm.jpg
fa46c89cf8f03662ad8e1cc499341f8c
af65f5f4c8f646ba5c21f7d1d91e308d9accd0c2
8890 F20101206_AABCYA peng_j_Page_22thm.jpg
c38e148bf330786ddbf08977fd839218
a926200dfcd0870f1af880b162da93c0cffa66a3
45690 F20101206_AABCTC peng_j_Page_57.pro
26ef94acad25bfc397b69ad315cf6f7d
ceeef26aa7e2626d3cf8ceb869038dbf0e7e4826
41824 F20101206_AABCOF peng_j_Page_68.jp2
09fadd1eb160d79b84d37753ef399d92
aa2269f64a24eb28de75a5f724c9fd5d5c25a3e9
95623 F20101206_AABCJI peng_j_Page_80.jp2
a7b6f54edf14d9ea8a4b20559b4d04fe
33d307c0e723e057279d386d1751e45d4defe7b6
24196 F20101206_AABDAB peng_j_Page_53.QC.jpg
64642c087698ea36562d444f69b62079
8d9fb72887c4a92bac6d05b4d5cbcd36a55f3f24
33879 F20101206_AABCYB peng_j_Page_22.QC.jpg
714cf290ff07e5b3c841a7a23d7d48f5
86ae181ff345240a86213839eefa635e180c3ba0
39537 F20101206_AABCTD peng_j_Page_58.pro
8f81e61835ac520146d5db3fed36d1ed
ba60c215556905ed64d2f74e9bde88dd7b52cddd
49441 F20101206_AABCOG peng_j_Page_69.jp2
befddea17806bf3a2242d5084002fc2e
c201922bde7f2d11c1dffeaec22afc6013ba66c2
1126 F20101206_AABCJJ peng_j_Page_40.txt
953a956720b55b5ad13abbd0578ed1ad
017dfe46af26f9e4bca4bd5135455a41facbfd41
6275 F20101206_AABDAC peng_j_Page_54thm.jpg
fc72747f4b22a0c7d410ced1117f4d96
ebefd990638777b4e31618f29c0799c43e716ec5
25831 F20101206_AABCTE peng_j_Page_59.pro
7cc645ba9b44be45c244422a2271dd09
72585f3fef9d4526b4fab107d8f1f3cd9843fb11
70091 F20101206_AABCOH peng_j_Page_70.jp2
4bfb2408585b7b319c36ca83c16d6694
0665d49de3afa8e85b3da0ff5af0f024359ab7eb
112230 F20101206_AABCJK peng_j_Page_56.jp2
a44d7ce5c3ab7479d5372f148856260b
731d0a6a1dcb156a77d5b38f0a94949afa716dee
19726 F20101206_AABDAD peng_j_Page_54.QC.jpg
84660ddab1aa7c35f270a4a9a61ef3cd
ac9a0759384378a3eb0c41e4a72f33244dfa92da
8790 F20101206_AABCYC peng_j_Page_23thm.jpg
d2395218c976709475dd5545d2e35299
b698c8ed1b88900b643fc5fea3b5fe276152c59e
51090 F20101206_AABCTF peng_j_Page_60.pro
a4ffb692c748c5f48d94b30f200fa072
2b06019b63790d9a90202b74406b7d0e3a55d1ea
67693 F20101206_AABCOI peng_j_Page_71.jp2
4ef8601fdf004db5da125c37174122f6
8fb986ce5127ffc6c403cd4372bd963058103b14
121255 F20101206_AABCJL UFE0021263_00001.xml FULL
7bf7bc9ad87808c770de485c05767978
7c51672fc38ce32fb027686c8961c55713ed6ba7
1946 F20101206_AABDAE peng_j_Page_55thm.jpg
ebd8744e898d4a3facaff409f47270a0
9c2bc32fa77d2826ba2b365c649398e07f08a098
35687 F20101206_AABCYD peng_j_Page_23.QC.jpg
70ef192e6765a6763c828fcb95e77566
e2e19214139fe1ead4bcb048f5a3190a67ae7f20
42443 F20101206_AABCTG peng_j_Page_61.pro
870495b6b7e1803ce4d62049e9b7922f
72754959ecf7f8697a7b5ab7f72231a177ed956e
68082 F20101206_AABCOJ peng_j_Page_72.jp2
4064a40b6580f81ad651b8a1dbf449b9
d7b9d801395e9a2dc1f1d0e8e64d1d41e90ce4a0
7362 F20101206_AABDAF peng_j_Page_55.QC.jpg
054fe1d1de50dca5a76c301c3471ce30
dc87c5d718e0caed6956c01040cf5e2441da13cf
28669 F20101206_AABCYE peng_j_Page_24.QC.jpg
50cd8001887567cd1e33b80c7e488b25
415a19e25002edd3b4c0a219c5c9f6fc1ea33cca
49260 F20101206_AABCTH peng_j_Page_62.pro
b5e500d170d9bacfd4da7f0d7e6063b6
aa365267addbb491138234f112cd8c20cd838cab
52865 F20101206_AABCOK peng_j_Page_73.jp2
db796091b9350840bad60a4896571cea
95eb227a141b904fdd080f58c59c65256f815112
8870 F20101206_AABDAG peng_j_Page_56thm.jpg
7fcfe356a8220d9ced1ed14867b0b318
a42bde3db9e87efa6e2f7037d93fd9a114da1f4e
6716 F20101206_AABCYF peng_j_Page_25thm.jpg
16cd6e15764a233934dc7727b984c433
01ab6bf548536f386868ea90c9c77fb5703295a7
41041 F20101206_AABCTI peng_j_Page_63.pro
0579f9d10801c67f9c9bf0a685a32efd
17be2cc50a829b67a75efdb9f67235ecb7d1e4b6
124535 F20101206_AABCOL peng_j_Page_74.jp2
661ab017de7a34b2bd90719c6e0e7df9
de26c4b3d3e89f50a8eff320e42c0391cc34d6f2
23201 F20101206_AABCJO peng_j_Page_01.jpg
0b9fb550f6f36099d7138a664394468f
a7118051d94a57d3d7f5594824f4243c7ece45c4
34514 F20101206_AABDAH peng_j_Page_56.QC.jpg
0cacac241fb38e8a39e17a85b4a9393d
3342d1c726a7f9bc2ef9b700558cd0d8af30a6ff
23394 F20101206_AABCYG peng_j_Page_25.QC.jpg
6716d0ffc55d5a2a77895dfc2fa23e98
8c7051bc67c1519125464417a9a5e28d4fb7e94b
36188 F20101206_AABCTJ peng_j_Page_64.pro
4dbe529b5499c24c2739fea4f712035c
c29291821437e091d0a5ff8ce81b344237df3e8d
51137 F20101206_AABCOM peng_j_Page_75.jp2
60a70832f6eb44f2249beefc340dcfee
6f6230d8132828f0f71254c893fc40402a349c37
3432 F20101206_AABCJP peng_j_Page_02.jpg
a9c4c1516136dc6c1d9377594311ccbb
c5131f1bbf16f367b9d19e283b92c2df7d88cdbf
8310 F20101206_AABDAI peng_j_Page_57thm.jpg
fec74ff6eb0284162631b1c00f97ec10
d61a16ea0ce20b787943e738273c13022588edea
7020 F20101206_AABCYH peng_j_Page_26thm.jpg
5d6c50d556c145f9484ee6d99663baaa
72cfe4b0fb7eb2e24df84b3d2f940672ae748279
39954 F20101206_AABCTK peng_j_Page_65.pro
f41281c4081db08ab4da57bee1efc0fd
54b9f0bbd66ca70fe38ee89126af775cd4297cf1
132071 F20101206_AABCON peng_j_Page_76.jp2
524161db5d183dbc9c4a38d9e9aeec6d
a4c6abd7023a265eb05d90a7dd16465926b5f579
3506 F20101206_AABCJQ peng_j_Page_03.jpg
f9ae4fc74ede55986f6f7bcd71c3df4e
e7b8f045e35d2c96f9106e46989a68ae8cffdb62
24389 F20101206_AABCYI peng_j_Page_26.QC.jpg
d3619e7f9bda37425bd01c9c618945ed
f850fb1e17fabb2207137b85e0ba512411251846
33092 F20101206_AABCTL peng_j_Page_67.pro
73324a60dd30b04302607e732439edf8
b3307eb5da4fcdd8bf57ebb2c7d080687ad04534
142598 F20101206_AABCOO peng_j_Page_78.jp2
0369b0f062fc3a9663c3c01f111af5f9
b7156e4c04bd1dd30e251fca0f9511b3e7e9cede
47197 F20101206_AABCJR peng_j_Page_04.jpg
1cc40ed8e6421fbe2b85c4a6c0723cbb
6fbfc9730380b5b08866cb482cf4db3230af82e8
30413 F20101206_AABDAJ peng_j_Page_57.QC.jpg
343cefec825a752a1e0859867df8ec3e
a8c1a81249c03f2160058a3c134e822a75e75df2
8586 F20101206_AABCYJ peng_j_Page_27thm.jpg
3685f5fbb528d9bd10be6b02a0bd82f3
5c628ab4b4b3da0bdb080d04f01224c00fd5664f
138649 F20101206_AABCOP peng_j_Page_79.jp2
647e36cbc8793e5c74de5c21e199b0e9
30caa5a775b0171fe71d808a124bc85962d5b9e5
92468 F20101206_AABCJS peng_j_Page_05.jpg
924fd41bad031d3c12cf3a9657f8eb0d
64ae7a29295bba09e0ed72c83b30d496024d5ae8
18418 F20101206_AABCTM peng_j_Page_68.pro
93c396cfabdfcaca9ff639c06b22f4e3
599c9ef7060da896423f21ffeaaf51dcf7bb98da
7708 F20101206_AABDAK peng_j_Page_58thm.jpg
3c1b3065464adf3722b76f13489e557a
2bea94b4f70000844b13ca9c199a930a99a04a0c
36398 F20101206_AABCYK peng_j_Page_27.QC.jpg
7ee2d6d129903962e92ebc47cb2d5d5f
4d4f40aa1ef9ab115acd72e5d11d541c2251f3ef
F20101206_AABCOQ peng_j_Page_01.tif
694c8f18fea82b8c85b76893661a8712
d350a12c8b894f0715384ca99901e2144566635b
24824 F20101206_AABCTN peng_j_Page_69.pro
e060cb343d6b2c73b46bc1a77a323a1d
dbbde12c477697a06d5d347a829120c9fb5477d6
29297 F20101206_AABDAL peng_j_Page_58.QC.jpg
d2aca792d6add5080fdc12572ede5d42
add5921e605f4693491a9348abb4dd431f0dfb42
29524 F20101206_AABCYL peng_j_Page_29.QC.jpg
3a1e7ba5ebe437d9f4b43aec2b396154
5c340a78f2eb5d5afd5ad565017e98a8bce43e05
F20101206_AABCOR peng_j_Page_02.tif
29031250676a2c80bb91a9e7668512e9
6a2e44536ed12edbf10df18b0dc103b39db3fac5
36533 F20101206_AABCJT peng_j_Page_06.jpg
b6d2988ecf65d84bd65fb5fb8a5941cd
1f6e43049140e870da549503fbf7074cbdc5371b
34074 F20101206_AABCTO peng_j_Page_70.pro
e19b914c0fea4efd7cbc76c2b96db346
2a12f8418318976aba8177b78f16c9b372f40d45
5627 F20101206_AABDAM peng_j_Page_59thm.jpg
55a9508ab77b7a9b103dc6ca65399c09
e1c7cba2a702b18acd041e100464f0760b7095c5
3708 F20101206_AABCYM peng_j_Page_30thm.jpg
92ace5694c353a48513e70b36a1a8ec5
e404e2e32b3e9a506821b2f7796f26507640e924
F20101206_AABCOS peng_j_Page_03.tif
83c26b77cf794b7bc94a3f9d90789122
399e62dfe86385f055b1f1e6b31e3b1bf2bfd823
46640 F20101206_AABCJU peng_j_Page_07.jpg
eea72ca173ab21e9865b2a58fb2e498d
065109bb07c57cb77e84252dc42c54229db58692
32139 F20101206_AABCTP peng_j_Page_71.pro
935baf104a2a598e154915b0373e4ee1
03898a1157f097b697211ab796d43f028ef71628
19063 F20101206_AABDAN peng_j_Page_59.QC.jpg
73507c4452e176bf7092e5a09a1e2d89
bf68715057430b47a5241b6580bfc4d4f5e92c91
13514 F20101206_AABCYN peng_j_Page_30.QC.jpg
6781cf1fffe38a355fd61b8c98646110
e436c82c5b8b2e9e7e4922b096e80651950f911d
F20101206_AABCOT peng_j_Page_04.tif
ad9b754143fce3ae4a9c6f1bc1526ef2
f170c82e09541df5426cb8816e447cf61bf65ed7
70888 F20101206_AABCJV peng_j_Page_08.jpg
d8e78f1a3f9565fd245195c574215d6d
0eb4f56e172161602004df6fe165de24b68d9cea
33892 F20101206_AABCTQ peng_j_Page_72.pro
2c03a4753457a37c82d296e9ec769990
9bf7680a4508955b9c581e914e75618baaf6b895
8825 F20101206_AABDAO peng_j_Page_60thm.jpg
9673108aa3221a198b8bf27874970a6c
4e02a506f81c682c7e43809f95e1d6e3377e5a82
8059 F20101206_AABCYO peng_j_Page_31thm.jpg
72072798986ca71dfa9cf38b8fcf4c5b
df75d5998fe1c31a0cfb4a45ed40669104664e57
25271604 F20101206_AABCOU peng_j_Page_05.tif
90c202f01a67f8e2a5abc80717d98a6c
f4270aa18735d33ddbd6198dc1a09ca153800a34
96579 F20101206_AABCJW peng_j_Page_09.jpg
699b3f60debf4d37dc1f9f8cf96e57ce
c1e1454adfd00950398ed017243977f884d73043
17439 F20101206_AABCTR peng_j_Page_73.pro
b57124fde09f7265ea0e908054772911
95522864fbef0b285a9c76822a5acab436271053
7989 F20101206_AABDAP peng_j_Page_61thm.jpg
cf53e100283241a0a4604c29ec240a9e
3aec516e4b08ff8e7d36766cd9bfa9e8188a72df
28840 F20101206_AABCYP peng_j_Page_31.QC.jpg
0d4d83f4dc9d2981f3aae91f3ee66cd0
ef2493eda83201dae770bed16e7565014ebbfde6
F20101206_AABCOV peng_j_Page_06.tif
fac6953a1061e74b2c5357bce1077c3a
cc773a467b4c39c53861f9f1c13cfbfe1d5907a3
91886 F20101206_AABCJX peng_j_Page_10.jpg
65b3ae591d93b8a2dca0da523281a372
d200ba762899a8e483a258e4297f5b35b3bfd4b6
23497 F20101206_AABCTS peng_j_Page_75.pro
4e0a47a0c30fe3a5473406f4f95576c9
8aa8e8f64956930b637f076cb01758f98baf879e
29327 F20101206_AABDAQ peng_j_Page_61.QC.jpg
257f964ca4d7f289a56615d3a6d1b0ca
126c7e40cafbd8e5807f4b2ee75246457b1ae95b
8006 F20101206_AABCYQ peng_j_Page_32thm.jpg
24198fe3942d47a932223df32239815d
3bafc13be79addb97c983fb39c9bce8edec6270d
F20101206_AABCOW peng_j_Page_07.tif
1201b7702196f2848b7a1b2016320a3d
7b01b1f2600cf0b3f80383b9fb2c09013cdd493b
68979 F20101206_AABCHA peng_j_Page_13.jpg
41f9e95103ff742985bab28a9a55b857
e275f1dc5c3caf6d9154368bbc8820fc08b0e199
104005 F20101206_AABCJY peng_j_Page_11.jpg
e101ff6724a9b0dad30d4c53ad7fdede
9a74c8b6f1dbead28b8b9a0e16b7589ca4e7a250
61315 F20101206_AABCTT peng_j_Page_76.pro
12d03cd4f63b6223419baec15d79c939
4d82e9e929f51729ff97afd47c22fa26df77431c
36369 F20101206_AABDAR peng_j_Page_62.QC.jpg
5ca0d641c7ee95ba206d9f3e0ab693b7
f97b5462b8eebb03b1fb7c2e3052c9f844a22f84
29632 F20101206_AABCYR peng_j_Page_32.QC.jpg
7bd07d971db26f7b290f76ee3c4ba6d5
705d4438f4acc8af012ade67d6a6cf58ac2aa7a2
F20101206_AABCOX peng_j_Page_08.tif
6428b68669d0aab590ca654f38212794
8a2747ffe55a7a76c0477332c2f01c5678cd64ff
36324 F20101206_AABCHB peng_j_Page_12.QC.jpg
9f32389a87e0eb7de8bc85001e3e573c
de66750f3526b9d6af79708a796145fb6224c8e9
110273 F20101206_AABCJZ peng_j_Page_12.jpg
6d862a18b090d762773b1bae59562563
df695b24ca574b696c70b1632bb711f42c380c62
57082 F20101206_AABCTU peng_j_Page_77.pro
064af876fff932e715aec331535028c1
2839e73f58215d31e0958bef53c8219d3a08ea93
26512 F20101206_AABDAS peng_j_Page_63.QC.jpg
452d0b4a3a9ce6592f69c03f23edcfb2
deb63ae33d5999b5574463bc6faebb474b2070e9
4281 F20101206_AABCYS peng_j_Page_33thm.jpg
66d49aeb7519f30255ec7214039a9f72
0604dbe5dac7113a2b6c3c83f448a793bf524ffd
47547 F20101206_AABCHC peng_j_Page_81.jp2
b8946b85feac3b35fa7bfe42b0680b18
4bffc19f9d267eb09ecefa013811cc85fc1db1fc
66622 F20101206_AABCTV peng_j_Page_78.pro
67d0c56d43e6cd9c853ff9030dd89edc
292854f034b1316bf9b5786d4dc9d2d9bec5e4c0
7042 F20101206_AABDAT peng_j_Page_64thm.jpg
9aff0745719096cb84a53a3d23404225
1457a16c854442b67dbd4364991897c26d66aea7
13401 F20101206_AABCYT peng_j_Page_33.QC.jpg
9a3ff678b2084e9af8139ed3d5f9833b
511fa33eac8779a70602055588d25c4d7520d8ee
F20101206_AABCOY peng_j_Page_09.tif
47171e9f0369454fa68c115f59a7cf1f
be8dd862f53c306796e519984f76e646f9bab6d3
1861 F20101206_AABCHD peng_j_Page_32.txt
a341f0844387ea464cd2a85e42c6d7df
b4564c838e77639216f88c82786b2cdc6fc72749
65175 F20101206_AABCTW peng_j_Page_79.pro
eb84766794acdb1ecbd2cbe10f69472b
fb63fd3c132e20d8dfcdb3523c5ffcf51a68443a
64767 F20101206_AABCMA peng_j_Page_72.jpg
4b5650c372d6d269b54e83894d0c6749
bb08f97f6049dcacb690d4811567c317affbaa24
24503 F20101206_AABDAU peng_j_Page_64.QC.jpg
7e9b8cdcec6e828c288b95b5fbc08920
affa13ca51a4cf37bef79b6d8d71581552afbbf8
6451 F20101206_AABCYU peng_j_Page_34thm.jpg
85c6f53d8e412d24ed08c0a3bb0111b7
8a76451b6d69cc23b802c6c653cbb81033805f1e
F20101206_AABCOZ peng_j_Page_11.tif
6b95813609fba8044fffeef490f17ecf
e6861194309daf0543ac604d5a5258212bf675da
29434 F20101206_AABCHE peng_j_Page_54.pro
4b6e2be01ccc6f90a5dae3fd13a04067
99710d456607059cb6cb0d4cf8429775c91d5dab
43062 F20101206_AABCTX peng_j_Page_80.pro
cfbc496c6f1a9a573ad625b304e3b4cc
6b2899c014d9431a3abe24ce72758bf4e0fe9336
118409 F20101206_AABCMB peng_j_Page_74.jpg
812fe8ec79c120f255ffc00083115335
990b5601ce15f46703393806992dfbe3b396b812
7370 F20101206_AABDAV peng_j_Page_65thm.jpg
4587b6e04d13aaec4cd9e58e567ab81a
93ce4de0a47b2d2813ba468d6a784b578a8d066b
22760 F20101206_AABCYV peng_j_Page_34.QC.jpg
5e135f5ceb1adf64ac148a084154ff8c
712aa62f0c14490ac8f02aea5485563f51984ced
F20101206_AABCHF peng_j_Page_31.tif
8256a0f495cce4f15c847108b226e9c8
accb0ee38df59a78cb2d818f3bbd318a900bb9f8
20154 F20101206_AABCTY peng_j_Page_81.pro
046439d6c4759237f666c72499340bc9
b82665be7ec565522aa7ff5c2da39118269c2070
49345 F20101206_AABCMC peng_j_Page_75.jpg
49a400f1c617e864209a785e418be757
b39774574cfd2d0ef79fdd0c649b55a617cc8403
28296 F20101206_AABDAW peng_j_Page_65.QC.jpg
dc3cb6e595c5819aa559a74a567d0921
a3aa4676030a208668b0c06cc4e9d383e1c89e30
7079 F20101206_AABCYW peng_j_Page_35thm.jpg
430b37d882306095ddeadc7f63e3c4fa
5ccbf614457ab183ec0e65251282231787f5f781
94259 F20101206_AABCHG peng_j_Page_31.jp2
e90d4419675dd467b36341d7d98f86ca
f18039faa8b5c7315fd3052dd4231b7e3a742a46
F20101206_AABCRA peng_j_Page_74.tif
f6fd97cd2decab361d5754a3932135bb
7ede623a7f5002b65cb6e2a5d6667bf888695949
410 F20101206_AABCTZ peng_j_Page_01.txt
c70747541a382d730906aea4e369ddbf
0234d34151558717f42c3b1077fae221683e9865
122322 F20101206_AABCMD peng_j_Page_76.jpg
205ca34440b04d861fe645e1a13f8e2a
22da2706a5a68cd158cd8e1a8156f389a1924bbf
8183 F20101206_AABDAX peng_j_Page_66thm.jpg
aeeecb80c23f5a5302ff9d0958127341
16594fb4d9006675b0311484e920b71bb4e761c1
26367 F20101206_AABCYX peng_j_Page_35.QC.jpg
0a7fc2d8555911d0d8feedd1944ad5ea
6457ee90ca794f192adb83de9ca516e1f2fd87ed
48641 F20101206_AABCHH peng_j_Page_42.pro
7534f4f81e8fce2187026cc5fc7d38e5
b0484f074295cdcfeb55f532d6d2b81a4df5e18c
F20101206_AABCRB peng_j_Page_75.tif
5e16913c552448c962151fecddaf991e
c931a851ec4f5241c295c15dcf357306d553b84e
113096 F20101206_AABCME peng_j_Page_77.jpg
5679d02a208d082175bd010116a1288e
b3f44c23844aeecb3bef82e222df3a88768ed72a
33570 F20101206_AABDAY peng_j_Page_66.QC.jpg
0fb74c6e1ce013e84b20b98558f1998a
d0160a787f933be32dadfdfac30b7c014fd27aed
7539 F20101206_AABCYY peng_j_Page_36thm.jpg
71fb66830a983efae08b4feeae049846
23031e36b061017ecd5e70ed1d6b71c1886d3ee9
92793 F20101206_AABCHI peng_j_Page_57.jpg
9fa981a21602df10fc561483da90084a
ca8429c63bcfd8e3883944b823ae353239819b29
F20101206_AABCRC peng_j_Page_76.tif
dfb9e9b750e440043616f2169297cb72
fc085f07ead699a61421ddaef2dfbe276735070d
89183 F20101206_AABCMF peng_j_Page_80.jpg
a977d1aba8d92453844720633b959789
f0df0854dfa297903e4d7f7cdeb08409eef1d320
23969 F20101206_AABDAZ peng_j_Page_67.QC.jpg
1bd7dade7157ba7851a7c1e37c361592
3910b3a4f098c1ea9d84a5e322ff2e830086df54
29552 F20101206_AABCYZ peng_j_Page_36.QC.jpg
d5c62fbd9b58ae9d758161210d74496d
a05512b646062660298d5ec1ea0d0a849c9bde31
1681 F20101206_AABCWA peng_j_Page_65.txt
a86cc46a34f1881ada9b2bbd4e6a7d9f
2203527ecece2773a4d2f37c87de54cc029e7fca
27296 F20101206_AABCHJ peng_j_Page_28.QC.jpg
6688b19d62764d540df124d8b88a8d53
e88bb697fe5a88398b44ca975fb9889a824dad0a
F20101206_AABCRD peng_j_Page_77.tif
52f728948b7c2b31b41fdf3c6365f746
64e5d991886965adfc322b0172a7c4a59be8d814
44525 F20101206_AABCMG peng_j_Page_81.jpg
e9ffd324f8abe8accf6f9101e83cb1fd
2f24e14c5030529cab0d6e458239891c1192136a
1855 F20101206_AABCWB peng_j_Page_66.txt
8bc5ffb65490cae17a7eab3018d89754
a912080e389e07a4ff6f6cb3968e1fb538c95876
20732 F20101206_AABCHK peng_j_Page_69.QC.jpg
bbab0bc548f123d762b8820507d6f9eb
7165cc9b31fda5f9a939201886528b07ab592a7f
F20101206_AABCRE peng_j_Page_78.tif
6e76381a0ef177fd80abe7f13c901f97
eb3e94640285d158e67fe32688b8f4452172b3a6
23004 F20101206_AABCMH peng_j_Page_01.jp2
cbab58f26b6a794e0889f79157740268
1246f2da8664afabd5f472cd541971dc89c78b97
1474 F20101206_AABCWC peng_j_Page_67.txt
1d42a0467459fa9b4c3564892d9a5a82
bba36bfcd6223e1848a46b94d67689ffd4b422f8
60113 F20101206_AABCHL peng_j_Page_74.pro
b140c58248d8d491cd72b842952a1dfe
e13034811abfe38a38caa5831d1edad2752ed923
F20101206_AABCRF peng_j_Page_79.tif
0700ea2b6ea722412b6141f6f6383b25
cfb00b353b785eb71c0c1ef7141551d4cd0396b4
4666 F20101206_AABCMI peng_j_Page_02.jp2
ce3f1ebdcb1ff2c8f1af03b4849854c2
ed07e9b8f219882ac58daaa1668d94e1db93a240
948 F20101206_AABCWD peng_j_Page_68.txt
b859a7a92f3a824502b9a2194af38a7b
0422ceb91333a4840cbfbd2eadf3a8d16b458b48
F20101206_AABCHM peng_j_Page_60.tif
276ea089aad0b440172951998ccfa592
a6e23c22e212f92150741ea3dfcf3529b6f5be86
F20101206_AABCRG peng_j_Page_80.tif
7c9002f8308fb7dfa6c466ed143ba66a
4bbbd33203050fcae257431609f444d2be188352
4552 F20101206_AABCMJ peng_j_Page_03.jp2
3d51a0fc9ef31cf6986a96a611020a8d
3b8d34aead1a7df7c14c1de8968d2e7c30ccac3a
1379 F20101206_AABCWE peng_j_Page_69.txt
073cb0d5b231aed3c163cd4c1ae4170d
b5396bea16ff8f57ea85cb6ef3ce6429f883506b
103771 F20101206_AABCHN peng_j_Page_09.jp2
97248de7466c598c8f0e00aa0b0673db
1e6ece298e85de5648925b08692d366eb4582441
F20101206_AABCRH peng_j_Page_81.tif
10057f233c1a2cd14d45e68f6193f79b
658b5ae0da532f40ae7449eddc91a5334d8b9668
48715 F20101206_AABCMK peng_j_Page_04.jp2
02f036f5fc62f0fcc74151c2b0bcc4d3
f1c0f92fa54da133a023846f058cd41ed5f1e6a7
1553 F20101206_AABCWF peng_j_Page_70.txt
bb002c4ad7a37c17b9f76c82624431a7
97483d55d02b79be3f8cd968096ea28fd0086071
91190 F20101206_AABCHO peng_j_Page_46.jp2
3b5f5393dff4f6694003f47f9eb5231f
92b4371c878d530936c095587d64a4dd7be9024e
671 F20101206_AABCRI peng_j_Page_02.pro
3272db13271a85c35f8c3b70fc0c0b63
2539da6ba5629bd2936ef6301a3b228929f0e573
1051986 F20101206_AABCML peng_j_Page_05.jp2
21d9d74f6a85bfcbfa90f480718bacc3
c9150a260261d92b53505923f51d1bc48877dad6
1526 F20101206_AABCWG peng_j_Page_71.txt
a56842c59d4f4d008be6fd93aac0c6e8
ad1e5d05a932575cdc71c5f96909bb406845de3f
20519 F20101206_AABCHP peng_j_Page_17.pro
b8ef9ff55c52d8f80bae6f49f4ddb2aa
bb812fc2b75eed775cf8c51f68a4f11ead0909c5
674 F20101206_AABCRJ peng_j_Page_03.pro
520230fa7d51b6e3fc8777cb1401940f
8594843af9ef1891fb8893b1f00ae622bfeb39c1
610157 F20101206_AABCMM peng_j_Page_06.jp2
be7108d1fcb31e4b5faa2d721fc3dad6
030b541f86b9166e05863c7ae520e2dd30cc86fa
1600 F20101206_AABCWH peng_j_Page_72.txt
11b7074ed78a465c94d71ecc9593eba6
0154b836820718339a2c1b301f6fe3b21f7c6ca2
1597 F20101206_AABCHQ peng_j_Page_24.txt
eee96e7b8727dc38eb23ac7da6bbbc47
013dda022c7b9ad81f96661c45d89cf35a208aa4
21157 F20101206_AABCRK peng_j_Page_04.pro
7d7c0f06ae560a5c38972dcbf92b43ed
791a509df8569ac6c0ff5afb79549c1c3689efcd
832664 F20101206_AABCMN peng_j_Page_07.jp2
1db8ef667e6a0390fff87d4360610834
09f9cc12e8aef83b77fe35b48012f6b262354aaf
1027 F20101206_AABCWI peng_j_Page_73.txt
2e4653bf12eaf1d5f48d1455efb86500
8d225914be366807db01364934db33258e7dd8e4
25380 F20101206_AABCRL peng_j_Page_06.pro
196ed931f16e33bd3635c23eee831fda
840b2aa907f0290a1cec1c41c595cf654a90f9dd
94389 F20101206_AABCMO peng_j_Page_10.jp2
9c3779a43f7e20cae88ab5a145d2eed0
170b5a85a02306a1d2a61944f7ff8fc28e6d872d
2424 F20101206_AABCWJ peng_j_Page_74.txt
bccbc049decfe67e7613202cf917f794
248e1f5b729eb43708c6610da61b67247e28df94
69731 F20101206_AABCHR peng_j_Page_67.jp2
dddceaf80cfaa01fda6e0215cfd8cf71
c64eb9a328860a59a0ae88d9013f04abf28b9f46
27552 F20101206_AABCRM peng_j_Page_07.pro
4b6ab8c56ca6f296f7735c94abd71354
c1f87eb03c3e9b1df4be364182e1e0354d5b83d3
112208 F20101206_AABCMP peng_j_Page_11.jp2
4a8e13b362c030a7accd820eab070205
b012064b2edf768e68f4294999e341692aef6f08
2470 F20101206_AABCWK peng_j_Page_76.txt
5afc3a627ba5b09b2523a9df4c5d46c8
72a780f54ad7a097a249c342598872e4a1631095
22121 F20101206_AABCHS peng_j_Page_08.QC.jpg
f5e27d21e61b9fe29ba06ea70dd4933d
73671a752813d9c035ea85c11828b39a4551ab20
32680 F20101206_AABCRN peng_j_Page_08.pro
5503c487aa23be303936afdb51a9ea0a
d8e74702e142ef44c02bdbeb5600828bd70e9a55
101270 F20101206_AABCMQ peng_j_Page_16.jp2
6a906d44c827a06c9a86aff1ae16c8d5
3b8921e714b2622693204cb9ceb63ec333c485f7
2299 F20101206_AABCWL peng_j_Page_77.txt
19b8abbd3ace66c9d8c4553325d94d90
d582850a701a1eb6085c0bcd898002420d847a9b
F20101206_AABCHT peng_j_Page_69.tif
c59438b05f16ed7fba472d1f5af32903
e6d7fb972a2b43749004c9f13da61e727ec52dee
47621 F20101206_AABCRO peng_j_Page_09.pro
6a82389daa8da4bfd02df9bd4a0f1ae7
594eb0a292568b06d3001454ea6292669abeef4e
43239 F20101206_AABCMR peng_j_Page_17.jp2
b619eca9060c9f9c1cbae8ff9c7ccb1a
4e2cc67392b2bb116b1f94fbffac57e7735e6dec
2674 F20101206_AABCWM peng_j_Page_78.txt
dd9d19ec2e2dea048f933b6592b653e6
20c94533cd49f54bb43f5d61a24cff4b65f283ca
F20101206_AABCHU peng_j_Page_53.tif
11a7c386094f3aa1f2ba8db0249f5974
104877ebbf81700c107591121865620645dca997
42732 F20101206_AABCRP peng_j_Page_10.pro
b74d9544ed68e3f5739f012d5cfde6cc
9551818b20e61cedc89f39fa63b31d6d03c9ef25
73666 F20101206_AABCMS peng_j_Page_18.jp2
7c851af595411cff1816fe371846ed33
c567aa8132a22ee9b1c41f9f44495e730c221f1c
2616 F20101206_AABCWN peng_j_Page_79.txt
e76939610e4f0fac73e374dbc038675c
38a3f348d4e868218277a73512495a901cdad7ad
1985 F20101206_AABCHV peng_j_Page_45.txt
45ecb59fca73419d7445f32143825ffd
81d22bfc0889004adcb890eea1590a3b216415c9
52789 F20101206_AABCRQ peng_j_Page_11.pro
b5731891652d0d1db260b2bd478c288e
26604aeca2e02f54c12141f2be64ce9b73170a67
104760 F20101206_AABCMT peng_j_Page_19.jp2
f809c0b2b1af6e60cf66aa587dedc8b2
6594cba47480ba5e37199416eacd70ec217763a4
1757 F20101206_AABCWO peng_j_Page_80.txt
08f8fa8acf4efbdc165eb2b9f5dfbec0
c8aab3ede20c057a1d0b40180bd325bb68f78521
7499 F20101206_AABCHW peng_j_Page_48thm.jpg
60b28a8759e9f190ebc2ca8af22a4479
77d37c0eb40ddc935c8059d1ed7213de26508dbe
56108 F20101206_AABCRR peng_j_Page_12.pro
ad06707a732d4b189631471cde175eef
ba6c0503126b099f58db43f8a43c1caa19411311
102784 F20101206_AABCMU peng_j_Page_22.jp2
1464b4e645e263c468b7e04ba90b5a0b
d3a17497a89e5b4ee60dcd67bc9802d660fc6f81
840 F20101206_AABCWP peng_j_Page_81.txt
df00c060605ac3f1da810e3238e678b2
f6e54a1541cd968c57bd56e715a5b390bc321063
31335 F20101206_AABCHX peng_j_Page_18.pro
827749543e1181a2416ffed94b9c4022
844d57ff7a3d7791885cee1fa16cbf68722576d1
33110 F20101206_AABCRS peng_j_Page_13.pro
6b69a4ce541677f843fb8f9e7601d54f
6c17ae519da5adf1df6c2177fc5f148a4c38968c
104332 F20101206_AABCMV peng_j_Page_23.jp2
e22d721c06e1f9e116e2c4052f52ce45
d347b673a049fe13814d17a158be50ffc2df3a83
363363 F20101206_AABCWQ peng_j.pdf
681c19de9442286913203988490e2843
c3844aff66e07d991ca20f91103b8168066dbc29
131663 F20101206_AABCHY peng_j_Page_78.jpg
c94d7a7cf6e7a517cf53e6a48967bd9f
6ed49658177a277d9f73bb3359c2e03d50d35aa9
43709 F20101206_AABCRT peng_j_Page_14.pro
bda8a2893ac8bb647ded4d1ba643e80f
82b3a2e74f9d434c8978a9c154b26a8d99f79e1d
1976 F20101206_AABCWR peng_j_Page_01thm.jpg
4080c13851cc225bd343f40f086e9393
b9e0ce08ed141d5f606b6da909783a587723db02
9256 F20101206_AABCHZ peng_j_Page_79thm.jpg
b8d1b3b400e8f1022256c722fa82c6d9
fe6e323204d3b1fa914a87e39abbad5f5052fbf8
43312 F20101206_AABCRU peng_j_Page_15.pro
e69a0440c62d01387c1c29b20f2d027d
b847bfb3901ca3061a8d22f4403c80588e3ec103
79177 F20101206_AABCMW peng_j_Page_24.jp2
effcabc3b9f9242b7d05a71b63f1adda
688e4c07d49a39cdd161dad6f62164512aa1883e
7435 F20101206_AABCWS peng_j_Page_01.QC.jpg
65ef5223628a8eae2b9bb310bcc22310
6b4f17f62a36f91bc97bb9c0c86d5949979d5823
49817 F20101206_AABCRV peng_j_Page_16.pro
c0f7db37c15ac9d0bab3e7d522c00dfa
a7adee4a55dd06e6c621e502dc85603c86cc3b32
75817 F20101206_AABCMX peng_j_Page_25.jp2
db98952d85be8b92c0ab6e7e18667ba3
7978862f5b496401fa7de09ec13626c8192f216e
460 F20101206_AABCWT peng_j_Page_02thm.jpg
6e2050fb4128754f0fea8dbd18868513
453dd681ba9681e26e6bb9b2b86043f845dd018d
50268 F20101206_AABCRW peng_j_Page_19.pro
7dc34f0bfe66837b9e5ce8b7cd551d2e
95a125b08b6891887f8356fa134622ddf0acbae0
86724 F20101206_AABCKA peng_j_Page_14.jpg
0e9aceeda416b8ed0c0e53c3935d3d7e
2f078a566af44655c4dd97f62790ebaefa63c22b
80015 F20101206_AABCMY peng_j_Page_28.jp2
87bae9d50a445d6ba1d0feb95b7444a6
30b025f5a358237420436ffc7bb51fbb29954d69
1050 F20101206_AABCWU peng_j_Page_02.QC.jpg
b4997d7e4ddc9075deb13a8c9cfdc3b0
c4383b477c8a554de40200bcda08a3d570959c75
50346 F20101206_AABCRX peng_j_Page_21.pro
b1b369c7cad0638f4326dfbada6d6e98
84049bada0d13b4c64678fb8533f63454d538e7a
84963 F20101206_AABCKB peng_j_Page_15.jpg
ce1ab266be9e0a1f03ae2011407ffc43
3687787352cea99f05c2919551e82ebb2b94e17f
82986 F20101206_AABCMZ peng_j_Page_29.jp2
78bcd6f7aa8c91637142091f61823544
ff3d54665a2a7c2b587f4e0593cfcc7f7b4a0ced
458 F20101206_AABCWV peng_j_Page_03thm.jpg
3926eb05326abc0ea80f0087e8d43f3d
f7a35a4f1e2a862ce4f4ef0b9a77cc60aa64ee5b
49698 F20101206_AABCRY peng_j_Page_22.pro
fd89849ef0664af228b957aeb2a556a4
2301fbfe82626db859a5e8ffafbc3c2121632e57
98802 F20101206_AABCKC peng_j_Page_16.jpg
ed3f365fa9abb984cedf931a3e6084fc
7adf6b0d3cfb7d0e2007d8fe94fa15f2e8dc3c88
1069 F20101206_AABCWW peng_j_Page_03.QC.jpg
030f0a86e718c55298fc886752781da0
304efdf2e9039bbea44f7dadb3c54ae2560490e4
50945 F20101206_AABCRZ peng_j_Page_23.pro
5b7c0fbedea21c3a77072f62ab87a632
86731a9fa6fbed5458873f1ef7c7ad9699383874
44315 F20101206_AABCKD peng_j_Page_17.jpg
ee9c73ec3eede3f3e3e4b685601f2e24
9c696f281ae275374fb82f316de2daf10799d01d
F20101206_AABCPA peng_j_Page_12.tif
b9efeba426b949485e6bf46369c00301
eac33d92098a69c6af3f251fe93854b3cfe9d6c9
3934 F20101206_AABCWX peng_j_Page_04thm.jpg
4dc27400f5dc46a12ccbd7f4bdc533cb
3b623f572c4fb95b6ab7380867df90120e8eefbe
70983 F20101206_AABCKE peng_j_Page_18.jpg
ad6a86d37a362547b18b167a2d5e517c
081876573e2bb1ff230eca4452f7044d1a6bca79
F20101206_AABCPB peng_j_Page_13.tif
54aaf8a47c1e18c337bb9208ec2bc3b4
3a06bb20a773bbe27ff51cff2306763d2569175e
15228 F20101206_AABCWY peng_j_Page_04.QC.jpg
190ec7fe454d091e860bd02987160eab
dec77e46b9198a274ed1f9f24c3fce67d78b5dc1
101429 F20101206_AABCKF peng_j_Page_19.jpg
0137704598c998d5d6a4439a11eb402a
46af0bdb54a5620e4b3bd823822045b0db0ab23b
F20101206_AABCPC peng_j_Page_14.tif
ad8fac316c8e96ca044c80558a36614c
e57e52dc2cc914ae0a9c9811dc6d663e642278e7
5665 F20101206_AABCWZ peng_j_Page_05thm.jpg
7ffb1b10334d23b7cf3dfd375b9ff3e7
e2c127e7dbae2dc207de2f8b4e479c8046ad76f3
93258 F20101206_AABCKG peng_j_Page_20.jpg
1a15d057d373400e9463521d3b6df4c8
53d4aedd3f3ef3104b09bb30de00ee03f7d72263
84 F20101206_AABCUA peng_j_Page_02.txt
2805087191d230ebeeaa9512e8c82c33
07d63c5cb9868aaaf604e5b42dca7a0b4c5a5513
F20101206_AABCPD peng_j_Page_15.tif
d8895cc4366370705384855afbfd25d2
8e70e8f7ae1b3db947683b48ecf1a1c03274740d
99777 F20101206_AABCKH peng_j_Page_21.jpg
4bd1ad258ad661c142f10065e5a8ac5d
62872bb14497c68eca67017fb96da05317e4f77c
82 F20101206_AABCUB peng_j_Page_03.txt
d885a6993770913388b25b40169c0eba
bbbeb1180d48f2c1e79ca26c69ccab0ceb1b0887
F20101206_AABCPE peng_j_Page_17.tif
a34d910d315bf1d1b22697f1bec60aeb
0d0ba4701ef1318de3c16b858e46c974bca0438e
4420 F20101206_AABDBA peng_j_Page_68thm.jpg
94c80d471cae436e02b7f9d36a6c5049
27b3bb9ded49bc351eefc298f0efcb27df97b9b0
7316 F20101206_AABCZA peng_j_Page_37thm.jpg
7a307710d0236e2ac285a93c680ec93e
bc631d0a7b85971828bd18a3f7e8a868dd399094
95945 F20101206_AABCKI peng_j_Page_22.jpg
d4d07c5c033bc582c6fc18d470689014
ca37e2dd65f872abf92951dce87f55927e4c7555
2735 F20101206_AABCUC peng_j_Page_05.txt
5c3c4ad8c1e8a4cd237160518c4f4bd9
6534decb65ff6b927b5246bbbbb2340a9eec7fd4
F20101206_AABCPF peng_j_Page_18.tif
12f242515e46f4953f40c09c6e6b028d
9d2919a399286cd2ba5f423a528f07168eb28a19
14997 F20101206_AABDBB peng_j_Page_68.QC.jpg
60f62f4c5673b5d8050d1339579812b9
50a5d5f60703b1736c4931eec73deb649aaafa48
7917 F20101206_AABCZB peng_j_Page_38thm.jpg
191b9c61e944098120778528e203ce10
b07328600d2e244549a844fd2bce11309de1be36
75284 F20101206_AABCKJ peng_j_Page_24.jpg
91e5190db8baac1b999b11721d20e1fb
95217c03d72f703e6cba8a3a2d4f05539842a846
1097 F20101206_AABCUD peng_j_Page_06.txt
69e2db32423d29fcb089957d6ee7ce4d
b642935afa56a034ce375bd6a8c9812b575259df
F20101206_AABCPG peng_j_Page_19.tif
851e5bd75d193e4243e95ba9bcaee4ef
520537532b47fd6a79bfe7d4ab614f556405f421
5613 F20101206_AABDBC peng_j_Page_69thm.jpg
21a94efc5bd409677416c034b437fb9e
538ddb709a71a2c8dd9dc2d71a20f0390c3e7be5
34010 F20101206_AABCZC peng_j_Page_38.QC.jpg
cace7ca62cd2a2cd8d9fd3a46657b75c
9a175c1dd51fc8d69986341bb1d24307cfc8a2b9
69568 F20101206_AABCKK peng_j_Page_25.jpg
b51799711820915332d6ce34c5bc148c
003df974ebeb57e7d667e4f3bec58629b688a73a
1139 F20101206_AABCUE peng_j_Page_07.txt
35bb5e4dc4824ad6abb2b7f558260427
a61e06484a562f512ab89e2a6bf5cff60848a48e
F20101206_AABCPH peng_j_Page_20.tif
efa70fe85aa24d2e07f5128ed9175221
c723f08f41d221970924832fb43f5e31df277e8f
23984 F20101206_AABDBD peng_j_Page_70.QC.jpg
4f2faed6a237b73017c59ce4316bcd99
30263f31298eb10dc41f559a1cbce0f50d11af69
73918 F20101206_AABCKL peng_j_Page_26.jpg
690e564a3a68c3caaeb0d263162f17e4
7cdbdc5a0e36dcfe8288c5918342535c9624a29f
1498 F20101206_AABCUF peng_j_Page_08.txt
d58a0ebe9179f800f142a85c9b64c20c
95eef853b36da1db63b91c9094045122b37e0a2a
F20101206_AABCPI peng_j_Page_21.tif
89ba1e5c108cadd40667ba97caf124d3
505b207e89c4297445f700c1f5a68e97c934494f
22662 F20101206_AABDBE peng_j_Page_71.QC.jpg
660cce5684a8eb6ceb723bc2f2de32be
ac74cd012623de0db8fea0bd66eee4f75a0b025b
7390 F20101206_AABCZD peng_j_Page_39thm.jpg
0160d70edbe9fd8be93dd8cb4e2bcefc
ad582d215a08ad620065c1d617bc07ffc4289f6a
101549 F20101206_AABCKM peng_j_Page_27.jpg
cc2e1ed7907cbfea72805b3bafe7b8e1
7523d9692b87d90b7f57d817ba314d0a869ca553
1972 F20101206_AABCUG peng_j_Page_09.txt
f91d4e750d137d21f3d8338f9c8105f4
0c437231ff246db2f1e35dbfd4c020ad8d6cebc2
F20101206_AABCPJ peng_j_Page_22.tif
6159bc6bf9e5d6134e5bc02168a7dc53
ad31a8e1972437896b323da2a27628570313e7ae
6136 F20101206_AABDBF peng_j_Page_72thm.jpg
e91a04dae9b9066d6ef4d40a39e80eb9
b2c2116655dc658d7d095c8c6e5cbacc65254ce2
25617 F20101206_AABCZE peng_j_Page_39.QC.jpg
52438595324f3f5782a0060056d7ced8
2124a1eb1e207eba321979711c1a815676bae918
81614 F20101206_AABCKN peng_j_Page_28.jpg
0184bc823acce25e67f8da40dce271f9
baa326f9213c7865253f3f17ae552d6935207692
1700 F20101206_AABCUH peng_j_Page_10.txt
2f0ef009ccb43b0a2b22c4a40d30fded
32f01ef44275ffb6f5c48ee9b8ef11ded38a2a01
F20101206_AABCPK peng_j_Page_23.tif
25a9320eb3b9fcac527b7d20efc1ef9c
caadfb5ef67f0d0f7d5606ac21175318ad6f802b
23234 F20101206_AABDBG peng_j_Page_72.QC.jpg
10ffb8588569c3be4f0cb33beac5c9f3
7823182bf13cbecea2a6b90060912169969e5c0a
16186 F20101206_AABCZF peng_j_Page_40.QC.jpg
a2c63ebf5fe357e5c8f63a1e9129bc86
6525eaa7676d7b38e3de92f8eac6312e5c663b30
43111 F20101206_AABCKO peng_j_Page_30.jpg
cc2ba978df312940200392d4f5726d7a
bb0d931fe827cc0902a31866bfb6f7215fc7bfce
2079 F20101206_AABCUI peng_j_Page_11.txt
9ff18c561365de0f2e02f373d4fb2a22
22dd947e81d90161bcdb1f0001ef400bfbe4044d
F20101206_AABCPL peng_j_Page_24.tif
2c25a07894628203c9bdb4fad4cc17c7
ce1bc8906c90d489ca6eac34067f4ab9c40dbfa4
4778 F20101206_AABDBH peng_j_Page_73thm.jpg
e48dbdd410945f89b8757c6745fc61ae
31b92379ca5f3bb1edb168ac42f6b23f51aba26c
26073 F20101206_AABCZG peng_j_Page_41.QC.jpg
5ae5f00fcd720b283b827392f884f76f
80f421f756c6640c91fe0b819c8bfe5dfc9e53dc
88231 F20101206_AABCKP peng_j_Page_31.jpg
816dc9d4ecc3b375472a79ca7d2c7908
d43da97e6f2599d22cd43debf505eff0614e1d73
2202 F20101206_AABCUJ peng_j_Page_12.txt
bd4c25910444596949b265705cc08ede
9b9b6ffc134925f2eceb5e5b3972816c13ed9a52
F20101206_AABCPM peng_j_Page_25.tif
b781db87f9a939ad97b3f7578c463b04
44b249df1d4d44011dda7b314d9a31e103e92d2c
17363 F20101206_AABDBI peng_j_Page_73.QC.jpg
c62b7ba697377a969e6592573338fbf0
9ed8d33c5fc2548c435f554093715774a32376c0
8458 F20101206_AABCZH peng_j_Page_42thm.jpg
681f8bed00b9eb9b32e16ad7d59e25cb
eaf6baaceb1aad25fb76e17698ee5bc841b193c7
90873 F20101206_AABCKQ peng_j_Page_32.jpg
66057bd2037f8295ff7cc6d448b7b0cc
743960b7b9e4b7470f078f73d250abe2106932ce
1323 F20101206_AABCUK peng_j_Page_13.txt
0f98ebe79e81fb62e3bbd942636dcd07
7640fa917d69bc5ede250544d8408a0bd6be6864
F20101206_AABCPN peng_j_Page_26.tif
6ab1a0a0250f6af8efe3657c304e2d01
c36263ffa7de6ad013f4033c81cf59603a27f585
8592 F20101206_AABDBJ peng_j_Page_74thm.jpg
ec86152c8a1608c256cf26f118fa5929
0c5bc8758510799fa1723d05e6b8c5ade7ab768d
33368 F20101206_AABCZI peng_j_Page_42.QC.jpg
f8efd5c82c3ef9813dce664c9a3163bb
2dd7da2437071bd7b081d7cc403b10b7a58b7d6a
39848 F20101206_AABCKR peng_j_Page_33.jpg
e9eef9fa0c120f079243a988ef4c8e54
5c61a4cc3b158fe61ba3979e79c962c62056c014
1756 F20101206_AABCUL peng_j_Page_15.txt
c26ba0f47a8565a5895d69161143485d
3060f40963d11885f1671129eda727816f60bad4
F20101206_AABCPO peng_j_Page_27.tif
259ea87d6315aa3d5fef3f4f8b1e8f54
ace3f464b6bc42449fd31fa10700e80d90c2f301
6644 F20101206_AABCZJ peng_j_Page_43thm.jpg
391811f183b015532d9847e4b07d4e12
5a0b181bbdaf7dfd253ae6231791a52399a1b9ab
62239 F20101206_AABCKS peng_j_Page_34.jpg
61394cf0f66d63d533f263326d11c21c
b36c6444c57a61f58b8d272f0beb4b836f8ed828
1988 F20101206_AABCUM peng_j_Page_16.txt
97ebfb5b4863e4fae10c6f365105af5e
908ce430934b32b5264b6ddb2c71fbae20e1420c
77731 F20101206_AABCFU peng_j_Page_48.jpg
f15ae24058b4fedd33a0ccc8a7683cbf
270362d43707544591f724946a1de33e8d4acd24
F20101206_AABCPP peng_j_Page_28.tif
5a490d0db2df4cdaaf03049dd46194b4
f8fccfeae93d51583423e239415bd80164219d8a
3761 F20101206_AABDBK peng_j_Page_75thm.jpg
968f6cc0bf13ef2b98f9f4ba64aa6710
229704b7a013910dd90de2285693f0045cc40e41
23772 F20101206_AABCZK peng_j_Page_43.QC.jpg
d8f60eccc561295f4860c9b018a03462
4878387039d574386f1321b072b95cce9768fccb
69193 F20101206_AABCKT peng_j_Page_35.jpg
676689eebf0f648751780bea0c24a061
5dbf10f55d6ba67bedd90cced0e20198e48d33f2
1302 F20101206_AABCUN peng_j_Page_17.txt
3300d5daa4d1bbf7c5ddd9afdc74dae5
905a35f986d7fc46d81bece4e842c4847fae300f
81265 F20101206_AABCFV peng_j_Page_29.jpg
92322cb1dc24481a07886b9bcc998bfa
d8155a4a1855f7f13773169945e240234737dc55
F20101206_AABCPQ peng_j_Page_29.tif
0cb4cc44cbbf7dd3608dc06a02c062fe
582f76d76971a1f4f8b2734817e3196f93c266d1
14844 F20101206_AABDBL peng_j_Page_75.QC.jpg
3bd1292b49e9334fbd6c0afe73c108de
78754b139c61a646eedd0940dba33dfba00ced2d
7853 F20101206_AABCZL peng_j_Page_44thm.jpg
1451f375d4aa8a8f8442cd4d956ba778
bd1a1aaa43aa46aa96284c9e56f9f8874fe780a4
1999 F20101206_AABCUO peng_j_Page_19.txt
6a6b93f0b1d037a91f9981997a5e0bee
e4cd54127f99272503f62586f78dc6b2b920dd09
7303 F20101206_AABCFW peng_j_Page_24thm.jpg
d53ae1420cb75327ec6c47bddea2ab74
6a416ffaeba6ab5aa1e880fb3e58fcb6dac82f89
F20101206_AABCPR peng_j_Page_30.tif
c03fb961e821ce526f83ae7c8d655fd5
00e575fcb07d7558b2d6916647febb12b2cda29c
8771 F20101206_AABDBM peng_j_Page_76thm.jpg
1e92f4261720f96ce00bb6801bfde0b1
85a199c7005060d75bfe4e9336e212f07c3b452d
24369 F20101206_AABCZM peng_j_Page_44.QC.jpg
71393005cec0652fcb30c470a6ae4dbe
82b32d75d87f6763c1120162ff687c2b2a15f37a
85736 F20101206_AABCKU peng_j_Page_36.jpg
7f377bd2b7265d6c79bac5c5090041c0
5a6769f107a1c5339d743473bd1ac20a24435b1c
1893 F20101206_AABCUP peng_j_Page_20.txt
b5893827ac820e78c2096c9de3057c69
8f76ebc9f0af98eae6b9875e58a1fd86191a1bee
86831 F20101206_AABCFX peng_j_Page_15.jp2
8aae7b4adbf555966bb5ff9216295876
7167a0ce2e24c4e2e5dcc272c300f4e057ac822c
F20101206_AABCPS peng_j_Page_32.tif
7665cc565f480678c8b69f29bd47bc18
3ea9a2f244c280eccdbc50021838dfe153d0dbf3
8591 F20101206_AABDBN peng_j_Page_77thm.jpg
ca3d76256fd0dcb5d210b72582604715
846dd5ca78c33ab4c7dfebc46f6b433e5b416f1c
8579 F20101206_AABCZN peng_j_Page_45thm.jpg
1b68cc297b892fd411907ecd59c9e098
da9881b8c8cf5686fe662826221e5e2f49a2216c
81311 F20101206_AABCKV peng_j_Page_37.jpg
bf93eefd5fdeccf5abba41aa5e4f9e28
07ac7367d400bc0c1c1f5c9dfc58de5f19295ca5
2019 F20101206_AABCUQ peng_j_Page_21.txt
571fd3366b61c9afc215c66c92d58d24
7966f019b04c9f81dc88249dbdba069addddca58
1670 F20101206_AABCFY peng_j_Page_37.txt
22b9296fa65b2e7f108a8bb6d7f0207c
ba658ae4cd4c7ef32cd18ff5a7ffafb362c5ef9b
F20101206_AABCPT peng_j_Page_33.tif
37e0cb5f254cacca0bf21dc2024c36f0
1c4d483a0b08ecf40b52239265ea83396c679245
33655 F20101206_AABDBO peng_j_Page_77.QC.jpg
083bb753b079e7a796a2d23c00d86eb3
d040816f513231b3afc9ddddb235416e9a5f7843
34365 F20101206_AABCZO peng_j_Page_45.QC.jpg
5ffc1e520780886349a404073e64d86b
54cb13e860e67fd33bce192c928ef41039916012
88560 F20101206_AABCKW peng_j_Page_38.jpg
cff0a808ce16139388c3fed7927d7ad2
a9f4fc38a5dd719e287e4651ea6456cbc9407e99
1986 F20101206_AABCUR peng_j_Page_22.txt
7127986e0aada5f3fce7b7ffc81ab24c
370c84128cdca1995c790a7e912e53f7be97af19
97492 F20101206_AABCFZ peng_j_Page_20.jp2
2960bd538e76f6a3190f1cf26f598156
91a603f098dae888a54e7c5313111437cf34b7c9
F20101206_AABCPU peng_j_Page_34.tif
507ab2a9aef65b51590e67fa88ea2f2b
2f0dd8b1bf764b691fbddcd94cf25307f4742ad1
38187 F20101206_AABDBP peng_j_Page_78.QC.jpg
3efed24fb50d233c84116911d26bde0b
67b29f28f90ca47e147ead7f407d82e9c3d4b845
8044 F20101206_AABCZP peng_j_Page_46thm.jpg
bfb0e2139c5be4d6d4ebac8fb02125c1
614ed454230d6e83fd532b30dca456adee880678
2012 F20101206_AABCUS peng_j_Page_23.txt
f3ac28fb46e2b2d174e802884d497c4d
afc0840991ce06eb57e48623a234ba54a65e7332
F20101206_AABCPV peng_j_Page_35.tif
d8cc1d8ed5ca5aa9f46b50f6839e0c2d
e12537bf45363cec57f81aab7687d7f541191818
72127 F20101206_AABCKX peng_j_Page_39.jpg
115c573e166b5e1e5cd03fed880f8671
dc98596c0180065452c2d4d5bcde807de8663811
37824 F20101206_AABDBQ peng_j_Page_79.QC.jpg
e998eb7e8c10e83e83f37fed6023c7e6
5ebd5c5fa82ea0f1c14d5b3d40c7ec8d15cd23ce
28896 F20101206_AABCZQ peng_j_Page_46.QC.jpg
d7dce33f8a368f3d4d02b1ab7e03057b
96c691cef7bdd6881c63a5973ca45f2848844e85
1485 F20101206_AABCUT peng_j_Page_25.txt
e345b4b2fdabe4c3b735eeef0659c593
5e182b0289132ef7291a750cd8dc223aff910105
F20101206_AABCPW peng_j_Page_36.tif
912ced7215c17601235886a9c069bc69
e98d7d6133ad7bef9a2a8c88a479629270abeb6e
8372 F20101206_AABCIA peng_j_Page_20thm.jpg
c3d2851d51b9da60d1a5dcce70e53959
59aa1f0dbd1e2fffd51647199d17c1a146319554
47883 F20101206_AABCKY peng_j_Page_40.jpg
ac318473cc5de70ee0a1581e91863f2b
3c771c399cf721bfc9de5af88a591c192cdfa8c4
6700 F20101206_AABDBR peng_j_Page_80thm.jpg
3a4c5fce8721e3706d78781f65e707e8
98cc0ed5306f9f1f853eb0bfb39a666cee303438
8727 F20101206_AABCZR peng_j_Page_47thm.jpg
85b736d14f88154d2de314eb9fd38c5d
7c774bd6784530ca7a0ae67b4b70dc0d6041f2fa
1777 F20101206_AABCUU peng_j_Page_26.txt
8ff48da015719e638d9f3b84a29c6df1
573d7fb7a539ee78614ae867053ee6d68471e7b6
F20101206_AABCPX peng_j_Page_37.tif
4cee46f16092563362fe15961cb597fb
bac7d6a7007d20f093f0e25702c88e4ca3244108
90961 F20101206_AABCIB peng_j_Page_14.jp2
c7bf929e92477f89b3c47ac83b0327a2
920ac949fd982af52292e7ef9c38b24775c4db5a
80011 F20101206_AABCKZ peng_j_Page_41.jpg
e396782e05e870f7883c45fe5c260f4c
8874fc2abc439a9123abd4c38aa2c15d23b34386
26942 F20101206_AABDBS peng_j_Page_80.QC.jpg
f79a1585e94f11b2e867bc55e847d160
ab70d878024891c71d9b463df6be88643a175de0
28258 F20101206_AABCZS peng_j_Page_47.QC.jpg
0fdaf6fa15da7c7efcc9eaa2c1ee6d24
df2033ac44a52cb57d6f1925807ddda72abe5d8d
2013 F20101206_AABCUV peng_j_Page_27.txt
4a7b6d80605631f8f5376c778c6629b3
5347c162a06eaba54cd12027f669f0adecaef7ba
F20101206_AABCPY peng_j_Page_38.tif
ea50f6f990f1dad510f8b39cf2d36256
b70440580382e39e7160b96234ffb3d4d22eca57
48635 F20101206_AABCIC peng_j_Page_51.pro
9f7db110dec91cb2f8846e10f25261fc
68c98645b3c31577867d39d8a696412a17803592
93915 F20101206_AABDBT UFE0021263_00001.mets
e58e03146b1def9b6e3843682e76fe8f
876c8e70a11550722fac4973b23d0fb0c4d355a9
27696 F20101206_AABCZT peng_j_Page_48.QC.jpg
08250414fae705562467525e4e9d7793
215139b67b438597ddec77a4b2f5a4ae4320ad62
1508 F20101206_AABCUW peng_j_Page_28.txt
25a95f595cdefbabb9078ca751afe5d5
68c5743417ab4af44c732442eb66fd352c31049a
46376 F20101206_AABCNA peng_j_Page_30.jp2
2c6429c47cc3ed23f584db15868745da
2eef732ac8faeec94693de72f6855bbfb674b47b
34528 F20101206_AABCID peng_j_Page_11.QC.jpg
0370d5d335785a7bb654256ca11888a1
bc4da8d5bc7c15c6fa4b3c658cdcc6ca5bba3d17
30409 F20101206_AABCZU peng_j_Page_49.QC.jpg
117d79a056119e9e7c915adc45c6902f
77ba53ae36cc4fba0ae13bebbb07086db35d39e4
1977 F20101206_AABCUX peng_j_Page_29.txt
b087b2dfd5f06cb2f1ae8106cb73a62c
159bff21ca17d8df33f426caa134e713c393c9a7
94930 F20101206_AABCNB peng_j_Page_32.jp2
8e5611082bbfd726ed9da4ab4acc51f5
2f371082d48dad1982c9af78102d9c42ab27ee30
F20101206_AABCPZ peng_j_Page_39.tif
9eb3b667990da394c0efc83d0b0f3150
314b713fba891a79645b4aa6cd4de00b03240a5a
1753 F20101206_AABCIE peng_j_Page_46.txt
1096c63318138c0ecf8f1cc127788518
1d3166f246e654a2bfb0c9d0fca427016cb9dfeb
8089 F20101206_AABCZV peng_j_Page_50thm.jpg
fa88c6316d31b4013370b19c6e65eb8f
3770d872f1399a685b91f47c1fa54cf9423410f0
825 F20101206_AABCUY peng_j_Page_30.txt
d2b7e6c81212ff6477306bc4ff3202f8
7ee4eb0ba23ba2fc35dd66e26c0334fefabb1c79
41146 F20101206_AABCNC peng_j_Page_33.jp2
1e1b445b05001783688b9db6046f45ee
7b6db6b90049306e7ce6b987182112b26b750c84
7864 F20101206_AABCIF peng_j_Page_15thm.jpg
a3a51021bd8f0abef02562a69ad4dc59
40c563503506d0d528ad4c65c6c243ae7d363839
30997 F20101206_AABCZW peng_j_Page_50.QC.jpg
b71b63c492fe39ad0fa89d7f03a32e0a
408000fc9f92faa5a8bc0cf327f74c331b3022ce
1862 F20101206_AABCUZ peng_j_Page_31.txt
a892b67a27442ac86e5688d407cc52c8
dd89c8b75aeab6a32c4e291cd6fa96c09e77b039
67080 F20101206_AABCND peng_j_Page_34.jp2
f59c67325130e186fc8676544262eb0b
2b43a384e6dec72e1ed4b49f6daee1320e52875e
F20101206_AABCIG peng_j_Page_44.tif
18efc4626bc04c18cf9b72528c6555db
e5a3e88713dfb6a6a6f3fa527579f92945d0a201
39096 F20101206_AABCSA peng_j_Page_24.pro
8d78ffd32a29cce599ca88f487905308
0e8cb679939596c795d9f4d1d452be73be4195de
8166 F20101206_AABCZX peng_j_Page_51thm.jpg
21ddd237e91c074d094e56592531cb55
7f82a4d3f8e595aa9870dc9b627866dfbbe06ef5
72247 F20101206_AABCNE peng_j_Page_35.jp2
d0346d0fa6275ccb5a7f12e00ad3c61e
8f436391a474a8498be5c0338134cea91ef14b7c
7013 F20101206_AABCIH peng_j_Page_49thm.jpg
ea46293d266a6f04cc1e79a95a1198f4
9d7b07991c6956eab9aee9ab1ca8de59c0bbf7d0
35704 F20101206_AABCSB peng_j_Page_25.pro
1c35b573bdbe59615baf5351d68bba64
9f330513fb14ca59a26bdbfffeca5f45a8cab02a
31392 F20101206_AABCZY peng_j_Page_51.QC.jpg
f39525b9f4ad16388f12a35c39b8c131
730847535f65724392ac8434cd11ac4af88ff2c8
23531 F20101206_AABCXA peng_j_Page_05.QC.jpg
1797b1a880ec5bef3208ed2bf42fae02
5314ba36e20c59fa3109bd5a7964ad201b27f1fd
89898 F20101206_AABCNF peng_j_Page_36.jp2
23c646489f6b87fbf6cbaf502cba7fe9
c8af37a292cb2e0d263e695834165ff4fd07ed23
6077 F20101206_AABCII peng_j_Page_67thm.jpg
2a96f0fc17b381f79edd9bbba7d5fe19
9948f6010409663501cb3cd0ceb53abb40ab0f29
39894 F20101206_AABCSC peng_j_Page_26.pro
fc31a2b0739585f193adfaa9b86a37b5
957c195bc631b0b636c5d1f0e9da0cd0f449489c
23297 F20101206_AABCZZ peng_j_Page_52.QC.jpg
b96bf6eb40cc4b5d285ad9e155a15ffc
bee95539bee052b6ee9919cc7d45a13f811c8edb
85444 F20101206_AABCNG peng_j_Page_37.jp2
b9f63433eac54f4090cd6234cb207547
655440c20d3375ee889d646a418171f621015f76
7242 F20101206_AABCIJ peng_j_Page_29thm.jpg
fb19266afa1fbcfb357400cb32de8fab
9d193513f5595aa2b3422150db17b8a1932d69f3
49806 F20101206_AABCSD peng_j_Page_27.pro
244d6574cabd77d341e4ead183369dc8
bd6ca446ee25aa963126b215c47055c29fb84993
3016 F20101206_AABCXB peng_j_Page_06thm.jpg
37eaae264e81d9c7a18941195f48e141
0182fab9622ded5eb31a5c299e3f74aba4fca44c
88207 F20101206_AABCNH peng_j_Page_38.jp2
1acfd5e3fe3d66ede5973847eac959b1
cad071c53450ecd23aebcd9010a874669b40a865
35656 F20101206_AABCIK peng_j_Page_76.QC.jpg
0dd8cadd5b3ff3bd1249e008421c5d37
0fcec33a4818c2893f4e2fac156e850f6cdfb986
35867 F20101206_AABCSE peng_j_Page_28.pro
d72cad4cec0e325069f02083f8bb72bd
503a948baa4c1d87987b9fc04071a3eeac824ed9
10214 F20101206_AABCXC peng_j_Page_06.QC.jpg
5170ffcf94e5f0d35605b4d2cb874275
9b81674b5bb010407151cb774662f44281ad6476
79433 F20101206_AABCNI peng_j_Page_39.jp2
de705b55782add5066495b1c3b263020
679a938e0a5c9cfa8e01a4dd0ce788194bff4b86
34482 F20101206_AABCIL peng_j_Page_21.QC.jpg
71fbc3c73cc06415fc2a3fbbcf8b3719
174059aa1f4efbaa19f305246f303fd8bcaed849
40840 F20101206_AABCSF peng_j_Page_29.pro
0d9b1468223b64463ebb1caeb2bb9a42
7ae1bb0fb21246755a782c01001182bc17393d30
3660 F20101206_AABCXD peng_j_Page_07thm.jpg
6875467df45c6c6bcd75ef7fedf4deef
2fe0839b01ad42b93827d095cae11eb78e432e2e
51349 F20101206_AABCNJ peng_j_Page_40.jp2
e9dadf955f22a19cfd130e0db9f7d765
cd7ba569a03a644af7c65db81c923d66df4b5070
1324 F20101206_AABCIM peng_j_Page_18.txt
8161d637cf1f4687893d121aca9f31b2
d37d90a1d332f3770a4ff3b8e81c9e29422ae74c
20604 F20101206_AABCSG peng_j_Page_30.pro
d5d3b82093d8c6bd4e2aac9cb96cb12f
f22f2e7303fca6b89ff934571929273746b57750
13742 F20101206_AABCXE peng_j_Page_07.QC.jpg
99b9dae8cf61f96acd8a5cd2066c536f
9589c6f9a75ff9bffff27b5c65ffaded35fe0f53
83967 F20101206_AABCNK peng_j_Page_41.jp2
bd8786c7bb45622823ec29f074a5dc8f
70412c0f3b20b487134115adb67eb52a8dec6823
99679 F20101206_AABCIN peng_j_Page_23.jpg
36c02f44a6415f8aeb7488dab683457a
1f519899fe143df3a6dfa2472b7b5ab6c1ba20bd
44433 F20101206_AABCSH peng_j_Page_31.pro
5d0e52fd41646c66106dabf4af2404ee
34c31380960567b4d364ea5e557716d1c6f099bc
7571 F20101206_AABCXF peng_j_Page_09thm.jpg
68f42b990c26f8727c7d891520327079
3be9ba6eda34175bcaa8ce6e96f8316777431f87
101106 F20101206_AABCNL peng_j_Page_42.jp2
a375113b5b6dce64caf5065dbfa7f5a2
8c46c99d7492136da25e0905489eb0d50079a851
106548 F20101206_AABCIO peng_j_Page_60.jp2
e2f2c47ed44b02d44e59acb9d281444e
59baed8fbbb5c2d2fe2da8041826b9c5a084a556
45558 F20101206_AABCSI peng_j_Page_32.pro
5c6bcb3f18b0da36e2c836ed684c82c9
ead7ab39179e9b86b28fd68bb2263411ec6d7351
31012 F20101206_AABCXG peng_j_Page_09.QC.jpg
9c13c5aeef274811b789739c55b14e9a
ba718f56bd92a9535ec19a3f318f12258a904c94
70702 F20101206_AABCNM peng_j_Page_43.jp2
83ee86f5fc17e2cc54df940ab0d1cb25
c2b74b81b14e15f4b87605ef6fad1ece9dc61cd3
99495 F20101206_AABCIP peng_j_Page_62.jpg
ebfb281f09da1a42f3377db59fa9272d
8ad8c0131d1c887d522786b5da442ef353e7f318
17262 F20101206_AABCSJ peng_j_Page_33.pro
18753658001342f90235c61868060fa4
f790cc88f6fe69c2af0758658087f33132c78794
8092 F20101206_AABCXH peng_j_Page_10thm.jpg
841e71a9204e37457f96cdac4ecea6e0
eda7402e4a59ed02ee339a5012a8a65109e16519
104753 F20101206_AABCNN peng_j_Page_45.jp2
7c863c548d3464a03faa01d3fc80f1f9
a7b9257698d876a765c87d2fb055e82648d14720
6923 F20101206_AABCIQ peng_j_Page_52thm.jpg
97a2be9a83a22cd4a67c85f1f18ea821
9d2240b139fb81d8842d9fcb9bd49262737d8a23
31500 F20101206_AABCSK peng_j_Page_34.pro
1a988a8b4951f8511d5ca0f57d16ece8
52ca611257d6925a6e5ac025c98b0d61acd538b4
30609 F20101206_AABCXI peng_j_Page_10.QC.jpg
9b295276b33087975b9162de35325555
4146d972b1f022003b20bb6a25c7fd1d712a9870
97352 F20101206_AABCNO peng_j_Page_47.jp2
d43ee5486f79f5722ecfd464e839329e
fa2079a50de75d574f6a4e5e8419bae49859138d
F20101206_AABCIR peng_j_Page_16.tif
87ad001ae6f8501f55fc9c207d7effd0
40d21b26f5dd3ad8a7d8fb57e8087f23c29382eb
43354 F20101206_AABCSL peng_j_Page_36.pro
434ddc7aaccb729dd1904d9c241de4ef
fea401d7f4955ebab09ff45e422d6aea625deeb0
8675 F20101206_AABCXJ peng_j_Page_11thm.jpg
fa3b9c8a95d18e7571883c60f6a2ed17
a9380dfe26be7ed4c9fe169de7a73ba32a87ccbe
82932 F20101206_AABCNP peng_j_Page_48.jp2
55a830fb980f43c0fdb1f484125d2cec
87df4a8d8d1d293b3f381b6a0cd28e0082e8b00f
40496 F20101206_AABCSM peng_j_Page_37.pro
bb8be8dbad38af800e3c0e2eb7b52e5a
2b61559a8a11abe87f5c924a57813776269c8fd3
9050 F20101206_AABCXK peng_j_Page_12thm.jpg
cf521be0b0f1a042c7180bd89a1613ba
284a87190e88cffc1dbdde605ec1cb34507c50b0
91534 F20101206_AABCNQ peng_j_Page_50.jp2
b01223b6fb9df5deefb317768aa2dc17
f1327f60992dc997794a041f46fcc47526d2f659
944 F20101206_AABCIS peng_j_Page_75.txt
5876b56bb8c3e2ff5bfa7f518f76e74d
10ff37749d6e8efd66e5838ac6c0471f45f2b5e9
45566 F20101206_AABCSN peng_j_Page_38.pro
c5f8aed49336478700f1b352f74ea5c3
1d7b270bb6e6cc4f6d9f1a86457ecc1e81d1b188
5511 F20101206_AABCXL peng_j_Page_13thm.jpg
934c8c38d83e97738c87f3766ca17e59
dc37327906b1cd461b17be4b718579983645896a
104524 F20101206_AABCNR peng_j_Page_51.jp2
886efd1085b7b538fbc5425c4bc5fbf5
70d78fde59b83adf1fece802a293dcc8f4816513
F20101206_AABCIT peng_j_Page_63.tif
b0e6fed07295e247d60d07731b619180
2f221f02a0a68b8a414ca5ff55981bcdd6e998ea
36696 F20101206_AABCSO peng_j_Page_39.pro
27163bbe0d5e842c7326a8814904b756
f2549be0401eda43da6975b167e55295aa0f849a
22680 F20101206_AABCXM peng_j_Page_13.QC.jpg
d614965f7c15cdc7670ecbd981188db2
faadee4cb05af231a0b8742579a3120e67fd7996
75989 F20101206_AABCNS peng_j_Page_52.jp2
63c7b671bf544ffbed973e5497551d4a
67f300b75fc64a62417370b05176ac4a3ac082d7
8712 F20101206_AABCIU peng_j_Page_62thm.jpg
28d457a36babe629456a457d89f460eb
a4f9b996bee2ca5e695a4cf6572ca46564d52743
22532 F20101206_AABCSP peng_j_Page_40.pro
8b32daeb8eb1f6cc174bf44c23ae303e
e0ca048628745cfb17edba4050d17abf6501de6b
7867 F20101206_AABCXN peng_j_Page_14thm.jpg
7f12e743e0079370f4a85bc69ff3b89a
057c437accb81f1d0bfde5a8959ade32800f0454
73477 F20101206_AABCNT peng_j_Page_53.jp2
44696c6681113cab05e665c4df22db0e
04a3f21808e749a382a1412ead590c7909e6b901
F20101206_AABCIV peng_j_Page_10.tif
df7f0fec081f0451601be0591f66d12f
34e17626aa7d0fce07652a28154f4bc0c830b678
40795 F20101206_AABCSQ peng_j_Page_41.pro
cfd85bc3f5b40b45abab4565cf4891f2
9a1da6356b9aeedd8d88e91d79b226cd537dc3d7
29484 F20101206_AABCXO peng_j_Page_14.QC.jpg
394fcb41793decf7da8fe00a9fd9dcf3
c15a80fad9944ab661056d9882d877c8a4f0c38b
104102 F20101206_AABCIW peng_j_Page_21.jp2
81a03d96aa943ea555207bdc28260217
fb2857f40ae28adb942ee0603032d457ffee5cda
36976 F20101206_AABCSR peng_j_Page_43.pro
370f659bd6f8d4439587cff400621d74
f2a3649a3bb630c28e3d075e85fb1c6da044fcbb
62196 F20101206_AABCNU peng_j_Page_54.jp2
a7177a6a3941778fa58bf2ff6c0b58e6
03610fe04607376f7265cbc1e1fce695223151b9
25659 F20101206_AABCXP peng_j_Page_15.QC.jpg
9741388df0ac60a24bd7f226244eb111
cfd9416208b2943a32ad7d9e8caa6c31e4e5eaa5
80766 F20101206_AABCIX peng_j_Page_26.jp2
8fcd08217bf9f725e765d10ca33d1bd1
31de76eacf97dceb7c52683541a7f3849bf373b7
38601 F20101206_AABCSS peng_j_Page_44.pro
40001052bd8bf0f9675728d6c95009cf
15cf64b8e683a42d1ea218831809cb9395567b24
21380 F20101206_AABCNV peng_j_Page_55.jp2
afa28811e76b4b714cc6a3e5fd37298a
ddab14844b7ea2fe7aa9f96fb927ca0c39565c38
8542 F20101206_AABCXQ peng_j_Page_16thm.jpg
340af1627254b22d548c7c0469f46fbb
6070be3768c9ee724b108410a728126e4db96b42
7134 F20101206_AABCGA peng_j_Page_01.pro
05c04c99b2f449cce3a7ed6cc1dfbe95
65f1f317cbac49392efa847e27ac1e1d238cdc5e
130398 F20101206_AABCIY peng_j_Page_79.jpg
f4aae011979989ef62f4494467a05497
be85fb610f555aba1ee605faf6cb78d5a5f12ddb
50132 F20101206_AABCST peng_j_Page_45.pro
0a01d7b97076c94a73a1099de36ae4af
677a43627ad0e712929e61136a78a1b9ab3e3e40
96786 F20101206_AABCNW peng_j_Page_57.jp2
e3bf1cb1a1984007bc035a9c693eb82b
aff043cef8faba946afb32fba56587ae6e1fc5fb
32259 F20101206_AABCXR peng_j_Page_16.QC.jpg
67f630d85f0df18a84672fe0463d8463
b86cbdde81f0da17a4720cdf1beb4cbdf83c92ab
F20101206_AABCGB peng_j_Page_62.tif
8a24862861527236753551247ad74f8c
57a098d9d9bd1b4e28b402604bf7036ed01a3917
47705 F20101206_AABCIZ peng_j_Page_20.pro
407501a7dc09b5ca77b39a7d63f307d4
237eedb233bc75b36b34306f37800ebec55d16fc
43200 F20101206_AABCSU peng_j_Page_46.pro
43e35256f0ea1f5a2925b95898417b0c
230d355b981ff17779666c8bf697ee7fa8effdc5
4520 F20101206_AABCXS peng_j_Page_17thm.jpg
ae21bbedda63a209a4e747a3ccb5dd2e
fb570dc5739c53573748ce361c817c2cb61f0865
F20101206_AABCGC peng_j_Page_56.tif
e109145d21b43a35aee2dcd90124ed10
9c70b5c2db645bcca065b751c4650381ae624108
46608 F20101206_AABCSV peng_j_Page_47.pro
e1ab298de2ff5e71b4aebfaa679a47a7
d9b326de3f1deffa96784d7085aba21604ecded8
84784 F20101206_AABCNX peng_j_Page_58.jp2
a1f49adf5761b1de4b8776e423b038db
3c0080b49b15fdfa8dd138066f0931e6cd2f85e5
16512 F20101206_AABCXT peng_j_Page_17.QC.jpg
306f42a96318ab74fd16fecad9a1e9cf
42fe7653a921fc0a8d2738a995c9f8496cc9948d
6115 F20101206_AABCGD peng_j_Page_71thm.jpg
ac4017e7e792292865edbf1656131462
acb20fc7c022e64646c371d831b373f5d608d3e5
36624 F20101206_AABCSW peng_j_Page_49.pro
af4d4fcb17033987ff04eaeb695b7e41
3174c319f61be21faa25b565c2e41c868a4841b1
96212 F20101206_AABCLA peng_j_Page_42.jpg
02ef4d67e5a32ea63cc904a554d72a53
afecdd8e3ef77f92cff4446c8e93996f4c9ca3bc
56124 F20101206_AABCNY peng_j_Page_59.jp2
7c16f0b7e3cb74f902042cd88a350793
52c6ff12cadd8d987649abb48cb9ac823bafd13d
6556 F20101206_AABCXU peng_j_Page_18thm.jpg
b14cd96c0687cc44b425a54d56652902
afbc097c78fcb9cfc28d208885f99ca7ff36d858
122695 F20101206_AABCGE peng_j_Page_77.jp2
f00790a011e70e2e78b403fadaf37d00
06858d6b0a23fdd0df3f1214d8953fe38f76a316
44226 F20101206_AABCSX peng_j_Page_50.pro
d831712b6286cb27c8a05df72c0d18a2
d48dca539cd1e70ecb4988c56a913ca1c6b1b41c
67642 F20101206_AABCLB peng_j_Page_43.jpg
8807babbddcdac95e1866e4a2a727dc1
5100c1e273a9a8c82dfa8fc77c715bba64f6a590
89125 F20101206_AABCNZ peng_j_Page_61.jp2
e31cf7c6a9fa61406768a108632988d6
636346bed00898be233206753a735a441a07c0a4
26157 F20101206_AABCXV peng_j_Page_18.QC.jpg
d1014aa40ed36b38346b6fdccd9d8441
1ca71637ef8b768c9dfbebd2bd3bf8bd19afc3b3
7425 F20101206_AABCGF peng_j_Page_28thm.jpg
147cfdbf529fbbeb0583839a11352727
2f24b6633fbb2664e6973bd0e90188bfe64f50fd
36133 F20101206_AABCSY peng_j_Page_52.pro
9fc658c710e387cc6b3a30204e1d28fe
d5fcb15c8276910a2ea72f68054fbaebdbacce31
78413 F20101206_AABCLC peng_j_Page_44.jpg
aca2e94a32a06ab4d138bde0b04ef457
833713ae533707b075762c438597849c5934c76c
8362 F20101206_AABCXW peng_j_Page_19thm.jpg
ed30d110f9bbc3e59c45bc05709cb3e6
8c58eb911d755e62ac8291b5a244482f99bda7ba
7468 F20101206_AABCGG peng_j_Page_63thm.jpg
c3eeccd260dfa44155691414b150d240
3711ff89d01e5040a9ed705c1dd9e15ccb2ce7c7
F20101206_AABCQA peng_j_Page_40.tif
b51ee5cf39dc8829fd4650948fa33e31
dad56607bbdc09b73e730623d25fd0ef1424fcf0
34238 F20101206_AABCSZ peng_j_Page_53.pro
2334ba081f242066c38a026777c9a701
4d60ba4daf44bcb1fefad7bd522ec55241e09e30
102208 F20101206_AABCLD peng_j_Page_45.jpg
05d1df9fad735696febdd41cd667de08
8be2f1c5130cb9b0261028873b6a23cda38c5929
34680 F20101206_AABCXX peng_j_Page_19.QC.jpg
8b2452939037fd15caa1d43174d51cf5
ad9d3c1e8c83e62929ab8e7d7e26a4babf2d62f2
F20101206_AABCQB peng_j_Page_41.tif
f34e6d00180c9cdd91bc4ae689dc5c6c
4a1be75fcc1693dbe3de870dba2487392602e446
89179 F20101206_AABCLE peng_j_Page_46.jpg
6529f3785e0f1cfa8bf7c38e42938064
5c01acdcd71003f7e59cbe8940477c211582aca4
38368 F20101206_AABCGH peng_j_Page_48.pro
18c16c28774438614d1049b0c6f078f7
410b5c409cad6b3a7d759f2486dd1ccee2800b3a
31720 F20101206_AABCXY peng_j_Page_20.QC.jpg
f95f87c10d554534ddb12ad5a8e00b57
0555f51877890c11e7e47ce5ddb33f7a2dca7c21
F20101206_AABCQC peng_j_Page_42.tif
373e9a39788873c5c5f779dd4d99b456
513ab7ed602ba8913267438c7539fe508a4b9851
95409 F20101206_AABCLF peng_j_Page_47.jpg
cb3e1d9ee8b8a7f9a4be54ab372b21fe
47ed27e62c7088bc8797ab3669724ee89679ae26
46396 F20101206_AABCGI peng_j_Page_66.pro
1099fd933f09bf36a678c2d907013603
9dd0033271561fed8247b26dd91349a2a6c04d3f
9069 F20101206_AABCXZ peng_j_Page_21thm.jpg
8c375c689e4e2353e6e0ef2ab6b9c3a3
78ddc17ef93d6a7714bd11c9f2d55531f03d8973
F20101206_AABCQD peng_j_Page_43.tif
537c152d78839b74bee36aff80998573
25a80ef29447bbdff946a3bae4c06e389266ebb8
77224 F20101206_AABCLG peng_j_Page_49.jpg
c95427080c372f63c1d75562d1c4007e
1778e3a9fca684f7f8dee2fb8f57823f553a6b83
1304 F20101206_AABCVA peng_j_Page_33.txt
fa17196ccbbbf73ffe8697381fc086a7
a05f1b5f0ff9b37ccbfad816deef56543629febc
1883 F20101206_AABCGJ peng_j_Page_41.txt
f59495e0d9459e57a6a656b44376028f
664b8dd0e114cf3176962f6a86f4d3ba29415b50
F20101206_AABCQE peng_j_Page_45.tif
1482fab79a5e14e1142b6758268c0959
ac5a294a7e6b68e41f0ace8bfe8e3d6a49633d6e
89550 F20101206_AABCLH peng_j_Page_50.jpg
3cc8fb4fa76a4a92970dfa142d3cafcc
787fbc056e54835113fd71195c0030177479cb1b
1786 F20101206_AABCVB peng_j_Page_35.txt
e3016bc3423e369471f613518b29a8ca
77f3786aac6a6f1133a97b14b83aded0bee2a482
1693 F20101206_AABCGK peng_j_Page_34.txt
867ccd55e342862564493b93f965afb3
36a7ad9d0c9556741d053169d5e16b6b79f6ed42
F20101206_AABCQF peng_j_Page_46.tif
64d932373465c319a879f2ae8f9f850a
e7aea42fadf0ababd5a714aa23de050fc1475a67
99932 F20101206_AABCLI peng_j_Page_51.jpg
eea3830b0d3a8bb04c78d67f300e05a2
6514fc9f1f34ada73ffdcffa33f0c12fd0e15c27
2059 F20101206_AABCVC peng_j_Page_36.txt
1c2b116f61bfb682cb3adc77632730f2
1aa2d9e944d71a61d624a0903131dffbeee23fe5
36411 F20101206_AABCGL peng_j_Page_60.QC.jpg
02c657c1f72f67d7a0181462360e1aad
3cb694362b312c87effe0a6fbe286afd8c666101
F20101206_AABCQG peng_j_Page_47.tif
6b83390f24792f31658a3f6de45ab546
d77fb2e3cd56ba3a35a6e22dc0e0a749c39c05f6
72170 F20101206_AABCLJ peng_j_Page_52.jpg
d5dddbcd829738d476cd172dca0896e9
ebe9f1e309d51d2fd55ad97bb74d4a338f23a3c2
1819 F20101206_AABCVD peng_j_Page_38.txt
ef5b87077279957263bb97de9c210baf
da000e8252a786f3ab00112ea3b2113f18c127d6
F20101206_AABCGM peng_j_Page_59.tif
fbfe07f5d663d04c43ddeb994a2fbd42
ea41ab4bd661acc014e2bde3eaac141bb5bcadf7
F20101206_AABCQH peng_j_Page_48.tif
3e0a3595b613775c1a5dba41503ba4c0
e32ed856ea2a5af866ab93f881ea678cdfe8f8c9
58284 F20101206_AABCLK peng_j_Page_54.jpg
8a0ca18d98fa3ae494be76343d54968f
692f96e724c1af1ea959596d2661cbc27a301428
1656 F20101206_AABCVE peng_j_Page_39.txt
8167cc38611395f77793da05c2fcb4e1
2e3eb43b6203af604fe75e478f0d96b787e12e24
66444 F20101206_AABCGN peng_j_Page_05.pro
3624d301c1d1a8076253644ae147ad73
91de017cdee5f61f24a9f7f9887dffee5020bd95
F20101206_AABCQI peng_j_Page_49.tif
bb81df26188180a3c0cd204730315e25
b53cf93fbadba53d9597d29e762898bd09169ef3
19034 F20101206_AABCLL peng_j_Page_55.jpg
7a6ed73ed5aeb922cb5945d486956f93
1887e236456c049e356c928ff7a9a50b797e556d
1935 F20101206_AABCVF peng_j_Page_42.txt
e98637a98c1ca09e4def9de8c4dc3234
fd5499e0a4bc543e6e4f7789e17df65ea925b266
5687 F20101206_AABCGO peng_j_Page_08thm.jpg
badff9a6ab99334f5cb783a99a694ff5
96a53f33cb1ae2cffc648f1cda09d89037f29b1d
F20101206_AABCVG peng_j_Page_43.txt
fd5f4a4ec3042391ae7c5a27119c8d57
ef97be7e8ac3fbcc020ea34bc29ee5c102a12576
F20101206_AABCQJ peng_j_Page_50.tif
eb03f4a920b9fad36d79515267722918
087b2e31ee245e0a9131d3a26913917e1a433db3
106957 F20101206_AABCLM peng_j_Page_56.jpg
54460989bf41eb7478152aca2d037cc5
b6e2010cbf5f02a33c9e55d31b86f498b3d6a7f8
117828 F20101206_AABCGP peng_j_Page_12.jp2
b3483b55e896f6d3e18256b4d49c9857
d2d56925e493dc1e6fa939594799dda034d120f5
1715 F20101206_AABCVH peng_j_Page_44.txt
846f1350011f1f025ccd618409d2f0b6
558f59fbdc03b22d7eec649b868f75f02884376c
F20101206_AABCQK peng_j_Page_51.tif
1467a7606836e082880242f5d5e48960
1dd531f32a50f3357a64ea5c021d4d7e71bbc8ab
83434 F20101206_AABCLN peng_j_Page_58.jpg
ca5284ed6825b89829a7ef3958478118
8f844f71bc86d49f9e297557759261e0ccb06c26
1849 F20101206_AABCVI peng_j_Page_47.txt
aa9b55639b3890e00b4218d3037c10e2
038ebaae8771eaa53e478e7408ea969016588649
F20101206_AABCQL peng_j_Page_52.tif
bd5e8f47424efa23c32f452dec65b370
87b78ae5b1e165b3da8cc79ad7d2bed61b89a3a8
55337 F20101206_AABCLO peng_j_Page_59.jpg
5d93dc1127c1057efbb363666816dd26
aca24ada102727166b5d0e023159a4c66293199b
34343 F20101206_AABCGQ peng_j_Page_35.pro
e498f83fad00e2ce4d26eece6fd44e6e
0dabc8570499506a793fb046eb4d00a9f86a0d15
1621 F20101206_AABCVJ peng_j_Page_48.txt
3e793cc91573d31abe8b88541efb3f90
866d7404fb74340a169304fe9701a68f8f1f9613
F20101206_AABCQM peng_j_Page_54.tif
f984b8444f99df8eb56d0d44fa2160ab
7263083361d00ef9378d49102bb7ddca3f27d879
102851 F20101206_AABCLP peng_j_Page_60.jpg
f7dd057be9db5cb462ecf5dd3cc6746d
cdaea461deed386a717028baabc18b9c32808494
72261 F20101206_AABCGR peng_j_Page_53.jpg
8f34f448eff5e0137e8f9fdf4fd6c298
0d575e7431f8725d73c249a7662aad314d27fe0c
F20101206_AABCVK peng_j_Page_49.txt
ee622c312cc117effba0cc57dde8a01b
9b651f704021edda7464c019b5877b435d821c89
F20101206_AABCQN peng_j_Page_55.tif
3e7bd1440cc1f3686d42b6e84672aa89
c2d1c2ee39a4d8b14aff7609ebfa21d88f160d8b
84983 F20101206_AABCLQ peng_j_Page_61.jpg
aef36a02da040a717f5fff91d0f5dc2f
2e382188e5b8377c03f4be28be46cd104612ea92
3749 F20101206_AABCGS peng_j_Page_81thm.jpg
bc50378a5f4c4eba4a083bec1777a754
649f32aba553dcf22c76443c15aa9c929c5cb729
F20101206_AABCVL peng_j_Page_50.txt
d8468776e301c15b40cadc162b0577e3
86dca3313d59ced3e979b840f6d95295ff70e90c
F20101206_AABCQO peng_j_Page_57.tif
07ca47298c526d1a417e69041c723935
492cbac59cb293bb6948c9c8ce907341bd8987c7
83500 F20101206_AABCLR peng_j_Page_63.jpg
31f35462bed8a5e020f150f27adeb5b5
4097548fe71ed7fa65d4b74077f9c30216b8a2ac
73515 F20101206_AABCGT peng_j_Page_08.jp2
3c800709ee2355ca51b8eca3e695eb96
9f76bf6f97601dfcf312626dc5099e6ab98322ab
F20101206_AABCVM peng_j_Page_51.txt
b69bafa3f6be78e034974890f4147a5e
2e37e1f28ac64939786e2ea45ea0350c03226b78
F20101206_AABCQP peng_j_Page_58.tif
76f25f41dcdf5c9ba7ad603db8db8ad3
b36fb5f1913ab8202cf6be794b3291a2f6c61d2c
74917 F20101206_AABCLS peng_j_Page_64.jpg
7409aaecca804d73c11ffb58ad0f62e6
c97825ee7130cc759646991c1c5f6b4ad7267085
47462 F20101206_AABCGU peng_j_Page_73.jpg
abc41e2d91ccae4ebfc6d17518c99400
7513b4ed02b8a4b312acdf3824fcec89ef3c939e
1570 F20101206_AABCVN peng_j_Page_52.txt
8e868f72432750e389b7a52da840146f
16fe2cdd0861c8509777ed0ac17dee79ebc0413b
82654 F20101206_AABCLT peng_j_Page_65.jpg
a1c5e3bf960f4a3ced8c727781bfc95c
c36b863ad6e0b46c3784af124a58234974dda36b
30178 F20101206_AABCGV peng_j_Page_37.QC.jpg
f214ffdecc6304032cec3a3c5039b7de
5566c3c346219e064cad541a8a5cbdb42abe4713
F20101206_AABCQQ peng_j_Page_61.tif
a3c378c332b3f865e6c6bfaa021fd46b
54aa2658d110252ccfd704f708dce7419891466b
1422 F20101206_AABCVO peng_j_Page_53.txt
3fb4f8d279c09403b1a9ee72acf9299a
3ab308ab9504a00ce58815ee2a7f28927b13318f
95188 F20101206_AABCLU peng_j_Page_66.jpg
e10c5f34f19b022a64fa3ee5f021a473
2ee09b566f507ed1eb6ad6e2154e7b90cb9f0ffa
14308 F20101206_AABCGW peng_j_Page_81.QC.jpg
20ace93047e7d6d149b3f7e2aace4750
d431193d7e63ae39d526b5952493b2df2f63f283
F20101206_AABCQR peng_j_Page_64.tif
091527d5e6df13d09c09a8bff0c9503c
0803b11dda9c820c9c4ddd0451ebfb325d3881f9







INPUT/OUTPUT CONTROL OF ASYNCHRONOUS MACHINES WITH RACES


By

JUN PENG













A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007



































(D2007 Jun Peng


































To Mom and Dad.









ACKNOWLEDGMENTS

I thank my advisor, Dr. Jacob Hammer, for his excellent, professional guidance and

support during my four years at the University of Florida. I thank Dr. John Schueller, Dr. Haniph

Latchman, and Dr. Janise McNair, for serving on my Ph.D. supervisory committee and for their

continuous help and advice.

I thank my officemates, Debraj Chakraborty and Niranjan Venkatraman, for interesting

and joyful discussions. I thank all my friends for their care and friendship.

Last but not least, I thank my brother and his wife for their understanding, support and love

throughout my school years. I thank my parents, Mingqing Peng and Mingxia Wu, whose

unceasing love and whole-hearted support made finishing this work possible. I thank my

husband, Zhipeng Liu, for his love and faith in me.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IST O F T A B L E S ..................................................................................................... . 6

LIST OF FIGURES .................................. .. ..... ..... ................. .7

A B S T R A C T ......... ....................... .................. .......................... ................ .. 8

1 INTRODUCTION ............... ............................ .................................9

2 TERMINOLOGY AND BACKGROUND ................................... ................................... 14

2.1 Asynchronous Sequential M achines.................................................... ...... ......... 14
2.2 Generalized M machines, States and Functions ...................................... ............... 21
2 .3 O b serve er............................. .................................................................. ............... 2 7

3 REACHABILITY OF A GENERALIZED MACHINE ....................... ........................... 31

3.1 G eneralized R eachability M atrix ......................................................... .....................31
3.2 Com m on-output G eneralized States................................................................... .... ..41
3.3 O utput Feedback Trajectory ................................................... ... ........ ............... 44
3.4 Prelim inary G generalized Skeleton M atrix ........................................... .....................51

4 MODEL MATCHING FOR INPUT/OUTPUT ASYNCHRONOUS MACHINES
W IT H R A C E S ...............................................................................56

4.1 M odel M watching Problem ............. ...................................................... ............... 56
4 .2 E existence of C controllers ......................................................................... ..................60
4.3 A Comprehensive Example of Controller Design..................................67

5 SUMMARY AND FUTURE WORK .............................................................................74

L IST O F R E F E R E N C E S ......... .. ............... ............................................................................76

B IO G R A PH IC A L SK E T C H .............................................................................. .....................81









LIST OF TABLES

Table page

2-1 Transition table of the machine ............................................. ............................ 17

2-2 Transition table of the m machine ............................................................................... ... 17

3-1 Stable state transition table of the machine g ............... ...................................33

3-2 Transition table of the m machine .......................................................... .....................43

3-3 T transition table of the m machine ......................................................... .....................43

4-1 Transition table of the machine ..............................................................................68

4-2 Stable state transition table of the machine g ............... ...................................68









LIST OF FIGURES


Figure page

1-1 Control configuration for the asynchronous machine Y .........................................10

2-1 State flow diagram of the m machine ........................................... .......................... 17

2-2 State flow diagram of the machine s ........... ...........................18

2-3 Control configuration for the closed-loop system ....................................................... 28

3-1 State flow diagram of the machine g ....................................... ................ 33

4-1 Equivalence of two asynchronous machines X' and ..................................... 63

4-2 State flow diagram of the m machine X' .............. ...................................... ............... 68

4-3 State transitions diagram of control unit F............................................. ............... 73

4-4 State transitions diagram of observer B ........................................ ........................ 73









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

INPUT/OUTPUT CONTROL OF ASYNCHRONOUS MACHINES WITH RACES
By

Jun Peng

August 2007

Chair: Jacob Hammer
Major: Electrical and Computer Engineering

The occurrence of races causes unpredictable and undesirable behavior in asynchronous

sequential machines. In the present work, traditional feedback control techniques are used to

control a race-afflicted machine, so as to turn it into a deterministic machine that matches a

desired model. Instead of replacing or redesigning the whole machine, I add an output feedback

controller to the original defective machine, and the controller eliminates the negative effects of

the critical races. The present work focuses on asynchronous sequential machines in which the

state of the machine is not provided as an output.

The results include the necessary and sufficient conditions for the existence of controllers

that eliminate the effects of a critical race, as well as algorithms for their design. The necessary

and sufficient conditions for the existence of controllers are presented in terms of certain matrix

inequalities.









CHAPTER 1
INTRODUCTION

Asynchronous sequential machines are digital logic circuits without synchronizing clock,

so they are also called clockless logic circuits. The lack of a synchronizing clock allows

asynchronous machines to operate much faster. In addition, there are practical applications, such

as parallel computation, where the underlying system is inherently asynchronous. Asynchronous

design techniques can also be used to achieve maximum efficiency in parallel computation (Cole

and Zajicek (1990), Higham and Schenk (1992), Nishimura (1995)). The design of asynchronous

machines has been an active area of research since mid 1950s (Huffman (1954, 1955)). Potential

difficulties, such as critical races and infinite cycles that may arise in the design of asynchronous

machines are discussed in the literature (Kohavi (1970), Unger (1959)).

Both critical races and infinite cycles are flaws in the operation of an asynchronous

sequential machine. In this dissertation, I will focus on asynchronous machines with critical

races. A critical race drives the machine to exhibit nonpredictable behavior, and it may be caused

by malfunctions, by design flaws, or by implementation flaws. Common practice is to rebuild a

machine that is afflicted by a critical race, and replace it with a race-free machine.

In the present work, I use traditional feedback control techniques to control a race-afflicted

machine, so as to turn it into a deterministic machine that matches a desired model. Instead of

replacing or redesigning a defective machine, I add an output feedback controller, and the

controller eliminates the effects of the critical races. The feedback controller turns the closed

loop system into a deterministic machine, and the closed loop system imitates the desired model

(Figure 1-1).


















Figure 1-1. Control configuration for the asynchronous machine Y

Here, Y is the machine being controlled and C is another asynchronous machine that

serves as an output feedback controller. We denote by YC the machine described by the closed

loop. The objective is to find a controller C for which the closed loop machine 2c exhibits

desirable behavior.

I represent the behavior desired for the closed loop system by an asynchronous machine

2', called a model. In these terms, the objective is to design a controller C for which the closed

loop machine 2c simulates the model 2'. Of course, the machine 2' that represents the desired

behavior is not afflicted by any critical races. Thus, by simulating the behavior of 2', the closed

loop system eliminates the ill effects of the critical races present in Y. The problem of designing

such a controller C is often referred to as the model-matching problem. The objective, then, is

to find necessary and sufficient conditions for the existence of the controller C that solves the

model matching problem. When such a controller exists, I also provide an algorithm for its

construction.

The literatures regarding the model-matching problem of asynchronous machines with

races has focused so far on asynchronous sequential machines in which the state is provided as

the output of the machines (input/state machines). In Murphy et al. (2002, 2003), the control of

asynchronous machines was discussed and state feedback controllers that eliminate the effects of

critical races in asynchronous machines were developed. In Venkatraman and Hammer (2004),









state feedback controllers were used to eliminate the effects of infinite cycles on asynchronous

machines; and in Geng and Hammer (2005), the problem of model matching with output

feedback controllers was considered for asynchronous machines with no critical races. In the

present work, I concentrate on the problem of designing output feedback controllers that

eliminate the effects of critical races in asynchronous machines. The problem of eliminating the

effects of critical races with output feedback requires the development of additional notions as

well as the development of new design algorithms for controllers, and these are the subjects of

our present discussion. To introduce these notions, I start with a brief review of some of the

underpinnings of the theory of asynchronous machines.

Unlike a synchronous machine, which is driven by clock pulses, an asynchronous machine

is driven by changes of its input variables. A stable state is a state at which the machine lingers

until a change occurs in one of its input variables. In general, the change of an input variable

causes an asynchronous machine to go through a succession of state transitions. If this

succession of transitions ends, then the final state reached by the machine is a stable state; the

states through which the machine passes during the succession are unstable states. Ideally, an

asynchronous machine passes through an unstable state in zero time. Thus, unstable states are not

noticeable to the user.

If an asynchronous machine has a succession of state transitions that does not terminate,

then the machine has an infinite cycle. Infinite cycles form another class of potential defects of

an asynchronous machine. The elimination of the effects of infinite cycles by the use of state

feedback was discussed in Venkatraman and Hammer (2004). Asynchronous machines with

infinite cycles are not discussed in this dissertation; I assume that all machines under

consideration have no infinite cycles.









To guaranty the proper behavior of an asynchronous machine, some care has to be

exercised during its operation. In particular, one has to avoid changing values of input variables

while the machine undergoes a succession of state transitions. If an input change occurs while

the machine is not in a stable state, then, due to asynchrony, it is not possible to predict the state

of the machine at the instant in which the input change occurs. As the response of the machine

depends on its state, this may result in an unpredictable response of the machine. In other words,

the response may vary depending on the specific state of the machine at the instant of the input

change. To avoid this situation, asynchronous machines are normally operated so as to guaranty

that input changes occur only while the machine is in a stable state. When this precaution is

taken, we say that the machine operates in fundamental mode. In this dissertation, all

asynchronous machines are operated in fundamental mode.

The development of necessary and sufficient conditions for the existence of a model

matching controller C and the algorithm for its construction depend on a certain generalized

concept of state, introduced in chapter 2 below. A generalized state describes a persistent state of

the machine X about which only partial information is available. More specifically, as X is an

input/output machine, it is not always possible to determine its current state from available

input/output data. A generalized state indicates a situation in which it is known that X is in a

stable state, but the exact state of X is not known; the machine can be in any one of a pre-

determined set of stable states. The generalized state allows us to use the partial information

available about the state of X to continue controlling the machine as best as possible toward the

goal of achieving model matching, while taking best advantage of the available information

about Y. The generalized state allows us to formalize in a concise and functional way the future

implications of uncertainties in the present state of the machine Y.









The notion of a generalized state was also used in Venkatraman and Hammer (2004) to

represent phenomena related to the presence of infinite cycles. In the present paper, I show that a

generalized notion of state can also be used to represent uncertainty in asynchronous machines

with critical races, in situations where the exact state of the machine is not known.

The mathematical background of our discussion is based on Eilenberg (1974). Studies

dealing with other aspects of the control of sequential machines can be found in Ramadge and

Wonham (1987) and in Thistle and Wonham (1994), where the theory of discrete event systems

is investigated; in Ozveren, Willsky, and Antsklis (1991), where stability issues of sequential

machines are analyzed; and in Hammer (1994, 1995, 1996a and b, 1997), Dibenedetto, Saldanha,

and Sangiovanni-Vincentelli (1994), Barrett and Lafortune (1998), where issues related to

control and model matching for sequential machines are considered. These discussions do not

take into consideration specialized issues related to the function of asynchronous machines, like

the issues of stable states, unstable states, and fundament mode operation. As a result, these

works refer mostly to the control of synchronous machines.









CHAPTER 2
TERMINOLOGY AND BACKGROUND

2.1 Asynchronous Sequential Machines

Definition 2-1 An asynchronous sequential machine X is defined by a sextuple (A, Y, X,

x0, f, h), where A, Y and X are nonempty finite sets: A is the input set, Y is the output set,

and X is the state set. x0 e X is the initial state of the machine. The partial function f: AxX ->

X is the state transition function (or recursion function) and the partial function h : AxX -> Y

is the output function. When the output function h does not depend on the input character (i.e.,

when h: X -> Y), the machine X is called a Moore machine in Moore (1956). +

Note that every asynchronous machine can be represented as a Moore machine. The

machine X operates according to a recursion form

Xk+1 =f(Xk,Uk),
yk h(xk), k= 0,1,2,... (2-1)

Where, k counts the steps of the machine E. The sequences xk, Uk, and Yk are the state

sequence, the input sequence and the output sequence, respectively. +

The machine E is an input/state machine if Y = X, or Yk = Xk for each step k > 0. When

the output is not the state, then the machine is an input/output machine. The present paper

focuses on input/output machines.

Definition 2-2 Let E be an asynchronous machine represented by sextuple (A, Y, X, x0,

f, h). A pair (x, u) E X x A is called a valid pair if the recursion function f is defined at it. If x

= f(x, u), then the combination (x, u) is a stable combination. +

Definition 2-3 Let (x, u) be a valid pair of the machine E = (A, Y, X, x0, f, h). If(x, u) is

not a stable combination, then the machine generates a chain of transitions x1 = f(x, u), x2 = f(1,









u), .... If the chain does not terminate, then the machine X contains an infinite cycle. If the

succession of transitions ends at a stable combination (xi, u), then xi is the next stable state of x

with the input character u. *

In the present work, I assume that none of our asynchronous machines possess infinite

cycles.

Definition 2-4 Let Y be an alphabet and let yi, ..., yq Y be a list of characters such that

yi+l yi for all i = 1, ..., q 1. Then, the burst of a string

y = YlYl ... y1 y22 .. Y2 YqYq" Yq is P(y) := yl y2 ( Yq-1 Yq

and PJ1(y) : = 2...yq-1, for q > 1; P31: = 0, for q = 1.

Let xlx2X3 ...xm be the string of states generated by the machine X from valid pair (x, u)

and xm is the next stable state. Then, the burst of the valid pair (x, u) is defined as

(xm, x, u) := P(h(x)h(xl)h(x2)... h(xm-l)h(xm)). *

Definition 2-5 Let p, and p2 be two strings of the alphabet A. As usual, p2 is a prefix of

p, if there is a string p3 such that p, = p2P3. We say that p2 is a strict prefix of p, if p3 0,

the empty string. +

For example, given three strings pl, p2, and p3: P1 = 1y1y2y3; P2 = y1yy2y3; and p3

y1y1y2y3y2y3. The string p, and p2 are each other's prefix string. Both p1 and p2 are strict

prefix strings of the string p3.

Definition 2-6 A state-input pair (r, v) for which the next stable state of the machine is

unpredictable is called a critical race pair, or, briefly, a critical race.

There may be more than one possible next stable states of the critical race pair (r, v). Let

us suppose there are m possible next stable states of (r, v). The set of all these states {r1, r2, ,









rm} is called the outcomes of the race. Correspondingly, there are m bursts for the critical race

pair (r, v), one for each possible outcome of the race. Let 3i be the burst generated by the

machine Y when the outcome of the race is ri. Then, we refer to the set 3(r, v) := {3(r1, r, v),

p(r2, r, v), ..., P(rm, r, v)} as the burst set of the critical race (r, v).

More details about races of asynchronous machines can be found in Kohavi (1970).

Definition 2-7 For a deterministic asynchronous machine Y = (A, Y, X, x0, f, h), let x' be

the next stable state of a valid pair (x, u). The stable recursion function s : X x A X of Y is

given by s(x, u) := x' for all valid pairs (x, u) E X x A. The stable state machine induced by Y

is represented by the sextuple Yi = (A, Y, X, x0, s, h).

For an asynchronous machine Y with a critical race pair (r, v), the stable recursion

function s has multiple values at the pair (r, v), say, s(r, v) := {r ..., rm}. Here, r r2, ..., r

are the outcomes of the critical race (r, v). +

Definition 2-8 Let Xr := {xi(), ..., xi(m)} be a set of states of the machine Y, and assume

they have a non-empty set U of common input characters. For an element u e U, let s[X, u]

be the set of all possible next stable states, where s is the stable transition function of the

machine Y. Let B(Xr,u) be the set of all bursts from Xr to s[Xr, u]. For each burst 3 e B(X,

u), let X(P) c s[Xr,u] be the set of all states x e s[Xr, u], i.e., the set of all states that can be

reached from Xr via the burst P. We refer to X(3) as a burst equivalent set for the burst

P with respect to input u. +

Note that the burst equivalent sets in s[Xr, u] may not be disjoint.

The following is an example to show how to obtain the stable state machine and transition

equivalent set for a given subset of its state set. Consider a machine Y with the input alphabet











A= {a, b, c}, the output alphabet Y= {0, 1, 2, and the state set X= {x1, x2, X3, 4 }. There is a


critical race pair (x1, c) in the machine. This machine X is depicted by a chart (Table 2-1) or a


figure (Figure 2-1). So is the stable state machine Yls of X (Table 2-2, Figure 2-2).


Table 2-1. Transition table of the machine X
a b c Y
x1 x1 x4 {x2, X3} 0
X2 X4 X1 X2 1
x2 x4 xI x 1
X3 X4 X3
x4 x' x4 2



a


Xi
























a b c Y
3 1













X3 X' -X3 1
b

Figure 2-1. State flow diagram of the machine Y,

Table 2-2. Transition table of the machine YEs

a b c Y
xl xl X4 fx2, X3} 0
X2 X1 X4 X2
X3 x X3 1
X4 x- X4 2

















c x' b a c







b

Figure 2-2. State flow diagram of the machine Es

Let Xr= {x1, x2, x3}, then U = {a, c}. Given u= a, the next stable states set is s[Xr, a]=

{x1}. The burst set is B(Xr, u) = {3P, 32} = {0, 020} and the two burst equivalent subsets of

s[Xr, a] are: S= {x1} and S2= {x1}. The state in S1 can be reached via burst P,=0 and the

state in S2 can be reached via burst 32=020, so S, and S2 are two burst equivalent sets.

From the above example, we notice that those states in a burst equivalent set of an

asynchronous machine cannot be distinguished from each other by an external observer. So, we

need a new method to deal with this kind of situation.

Definition 2-9 Let Xr := xi(), ..., xi(m)} be a set of states of the machine Y, assume they

have a non-empty set U of common input characters, and let u e U be a character. The

asynchronous machine Y = (A, Y, X, x0, f, h) is detectable at the set pair (Xr, u) if it is possible

to determine from input/output data whether all outcomes s[Xr, u] have reached their next stable

state; if so, the set of transitions from (Xr, u) to s[Xr, u] is called a stable and detectable

transition set. *









It was shown in Geng and Hammer (2005) that a stable transition is detectable if and only

if its burst switches output characters in its last step. When dealing with detectability of sets of

states, the situation is somewhat more complicated. The outcomes of a state-set-input pair (X,

u) form a set of states s[Xr, u] and all bursts from the set Xr to s[Xr,u] form a set of bursts

B(Xr, u). Even if every burst in B(Xr, u) is detectable individually, it is still possible that one

cannot determine whether the machine has reached its next stable state. For instance, consider

two bursts in B(Xr, u): 3(xl, Xr, u) and P(x2, Xr, u). The burst 3(xl, Xr, u) is a strict prefix of the

burst P(x2, Xr, u), say 3(xl, Xr, u) = y1y2y3 and P(x2 u) = yy2y3y4yy3. Clearly, then, at the

point where the burst 3(x, Xr, u) ends, it is not possible to tell whether the machine has reached

its next stable combination, as the machine might actually be on its way to the state x2. This

discussion leads us to the following statement.

Proposition 2-10 Let Xr := {xi(), ..., xi()} be a set of states of the machine Y, and

assume they have a non-empty set U of common input characters. For an element u e U, let

s[Xr, u] be the set of all possible next stable states, where s is the stable transition function of

the machine Y. Let B(Xr, u) be the set of all possible bursts generated by the pair (Xr, u). The

asynchronous machine X is detectable at the set pair (Xr, u) if and only if the following

conditions are satisfied:

(a) P31 P for all P e B(Xr, u);

(b) P is not a strict prefix of 3' for any 3, 3V' e (Xr, u).

Proof The first condition has been proved in Geng and Hammer (2005). Let us examine

the second condition. The first condition guarantees the detectability of the end of each burst in

B(Xr, u), so the only confusion is from other bursts in the set B(Xr, u). Consider two bursts Pi, P3









e B(Xr, u), where Pi leads to the state x', while Pj leads to the state xi. Assume first that Y is

detectable at the pair (Xr, u). By contradiction, assume that Pi is a strict prefix of Pj, i.e.,

Pi = YY2..Yk-lYk

and

Pj = YlY2 .. Yk-YkYk+l .. y, where, Yk Yk-1 and y, # y,.

Once the change from yk-1 to Yk is observed, it is not possible to determine whether the

machine Y has reached the next stable state x' or whether it is still in the transition to next

stable state xJ. Thus the machine Y cannot be detectable at (Xr, u), a contradiction. This shows

that condition (b) must be valid whenever Y is detectible at (X, u).

Conversely, assume that conditions (a) and (b) are both valid, and let P3 B(Xr, u) be a

burst, and let X' c s[Xr, u] be the set of all states to which the burst 3 leads. Now, since 3 is

not a strict prefix of any other burst in B(Xr, u), it follows that, at the end of the burst 3, the

machine Y must be at one of the states of the set X', say the state x' (i.e., Y cannot be on its

way to other states). Furthermore, since P_ P13, the end of the burst 3 can be determined, and

whence it can be determined that Y has reached a stable combination with a state of X' (note

that it cannot be determined from the burst which state of X' has been reached). This completes

our proof.

For example, consider the machine Y with transition table of Table 2-1, let Xr= {x, x2},

then U = {a, b, c). Let us check the detectability of the combination (Xr, a). The next stable

states set s[Xr, a] = {x4}. The burst from the state x1 to the state x4 is P = 02 and the burst

from the state x2 to the state x4 is P2= 102. The burst set P(x4, Xr, a)= (P, P2}. Since the burst

P, is not a strict prefix of P,, and vice versa, the transition from (Xr, a) to x4 is detectable.









2.2 Generalized Machines, States and Functions

The next notion is central to our discussion. It is sometimes convenient to consider certain

sets of states of a machine as one quantity. This is convenient, for example, in cases where the

available data at a certain point in time does not permit us to distinguish between these states.

This leads us to the following notion of a generalized machine.

Definition 2-11 Let Y be a machine with the state set X and input set A, let S(3) be a

burst equivalent set with respect to u of the machine Y containing more than one state, and let

E be a set disjoint from X, and let 0 : P(X) -> E be a function. Associate with S(3) the

element xb := ((S(3)); we call xb a burst state. The set E is then called the set of potential

burst states and D is called the burst state assignment function. Let A be the set of all common

input characters of the states in S(P). Then, the set of all valid pairs of xb is given by {( xb, a) :

a e A}. The set A is also called the valid input set of the burst state xb. Let Xb c E be the set


of all burst states of the machine Y. The generalized state set X of Y is the union X U Xb. The

burst equivalent set S(3) represented by a burst state xb is also recorded as S(xb). *

Let s = (A, Y, X, xo, s, h) be the stable state machine of an asynchronous machine Y.

Let Xr := {xi(), ..., xi(m)} c X be a set of states of the machine Y, and assume they have a non-

empty set U of common input characters. For an element u e U, let s[Xr, u] be the set of all

possible next stable states. Let B(Xr, u) = { 31, 2, ..., P, be the set of all possible bursts

generated by the transition (XV, u) -> s[X, u], and let S(P3), S(32), ..., S(P1) be the burst

equivalent sets in s[Xr, u]. For each i = 1, 2, ..., ?, we distinguish between two cases:

1) The set S(13) contains a single state x e X. Then, we identify S(13) with the state x.









2) The set S(13) contains more than one state. Then, we associate with S(13) a burst

state, which represents the fact that these states are indistinguishable in this transition.

For the second case, let Ai be the set of all common input characters of S(Pi), i = 1, 2, ..

L. Note that Ai cannot be empty, since at least Ai contains the element u.

Definition 2-12 Let Y = (A, Y, X, x0, f, h) be an asynchronous machine with a generalized


state set X = Xb u X, where X is the regular state set of Y and Xb is the burst state set of the


machine Y. We build now a generalized stable transition function s : XxA P(X) as follows:

1) For all states x e X and all input characters u e A, set sg(x, u) := s(x, u).

2) For a burst state x e Xb, let U(x) c A be the set of all input characters that form valid

pairs with x. Let S(x) be the burst equivalent set represented by the burst state x. For an input

character a e U(x), let A1, A2, ..., Am be the set of all burst equivalent subsets of s[S(x), a]. If

A' contains more than one state, then let x' be the burst state associated with the set A';

otherwise A' is represented by its only state x', i = 1, ..., m. Then, sg(x, a) := {x1, ..., xm}. *

Note that the next stable states of a machine can be a combination of burst states and

regular states.

In the next discussion, I will give a more specific algorithm to build the generalized stable

transition function. As a practical process, the algorithm should avoid getting involved into

infinite loops. Thus, before the construction we need to make sure about two issues: a) the

process of building the generalized stable transition function includes finite steps; b) there is no

infinite cycles created in the construction. Since every burst equivalent set S(3) is a subset of

the state set X, for an asynchronous machine Y with n regular states, the maximum number of









subsets in X is 2". Hence, the number of burst equivalent sets is equal or less than 2".

Namely, the number of burst states generated in the machine Y is finite. Then the first

requirement is guaranteed. In the previous discussion, we have excluded asynchronous machine

with infinite cycles. So, any transition starting from a regular state of a machine in this paper

ends at the next stable states. Similarly, under the definitions of the generalized state and

generalized stable transition function, for each valid state-input pair there is one or more next

stable states. Hence, each transition starting from a generalized state ends at the next stable

states, i.e., no infinite cycles will be created in the process of defining the generalized stable

transition function s9.

Consider an asynchronous machine Y = (A, Y, X, xo, f, h) with stable state machine s


= (A, Y, X, xo, s, h). Let X = Xb u X be the generalized state set and let Xb:= {1, 2, ...,

be the burst state set of Y. For every burst state c e Xb, let Ab := {a, a2, ..., ag(c)} be the valid

input set of ,c. For every valid pair (, a), e Xb and a e Ab, let S(,) be the set of regular

states represented by and let s[S(,), a) be the set of all possible next stable states of [S(,),

a]. Assume that 2s have p critical races (r v1), (r2, v2), ..., (r, Vp), and let T(ri, vi) := {ri, ri,

.. rm(i)} c X be the set of all outcomes of the critical race (ri, vi), i =1,..., p. We build the

generalized stable transition function sg with the following algorithm.

Algorithm 2-13 Consider an asynchronous machine Y = (A, Y, X, x0, f, h) with stable


state machine Xs = (A, Y, X, xo, s, h). Let X = Xb u X be the generalized state set, where Xb

is the burst state set and X is the regular state set. Assume that is has p critical races (r1, v,),

(r2, v2), ..., (rp, Vp), and let T(ri, vi) := {ri, rt, of the ..., rm(i)} c X be the set of all outcomes









critical race (ri, vi), i =1,..., p.

For every state x e X and u e A, if s(x, u) is a single state, then set sg(x, u) := s(x, u).

Set Xb := 0 and let K := (r1, vl), (r2, v2), ..., (r, vp)} be the set of all critical race pairs of the

machine X. Set i := 1 and run the following steps:

Step 1:

a) Consider the i-th element (ri, vi) of the set K. If i < p, then let Xi:= {r{, ri, ..., rmi)}

be the outcomes of the critical race (ri, vi).

b) Ifi > p, then the i-th element (ri, vi) of the set K is a burst-state-input pair created in

Step 3 of a previous cycle of the algorithm. Let S(ri) be the state set associated with the burst

state ri. Let Xi := s[S(ri), vi] be the set of all possible next stable states of the set of states S(ri)

with the input character vi.

Step 2:

Set j = 0. Partition the set Xi into its burst equivalent subsets T1, T2, ..., Tt with respect

to the input character vi, and denote by T := {T1, T2, ..., Tjt the corresponding class of subsets.

Let Z be the set consisting of all subsets Tj that contain a single state; if there are no such

subsets in T, then set Z := 0. Denote by S := T \ Z the corresponding difference set. If S' =

0, then set k := 0 and go to Step 4. Otherwise, Let S S, ..., Sk be the members of S'.

Step 3 :

Set j:=j+l and check the set S' as follows. Let E be the set of potential burst states and

let 0 : P(X) E be the burst state assignment function.

If O(SJ) V Xb, then proceed as follows; otherwise, go to b).









Add the burst state x := O(S) to Xb, i.e., set Xb:= Xb U x'.


Let A, := {u1, u2, ..., Ug(j)} be the valid input set of the burst state x'. Let r := #K be the

number of elements of the set K. Add to K the elements (r +, v +) := (xJ, u), ac = 1, ..., g(i,j).

Add the burst state O(S) to Z.

If j < k, then go back to Step 3.

Step 4.

Set sg(xi, ui) := Z.

Step 5.

If i < #K, then set i = i + 1 and go back to step 1. Otherwise, terminate the algorithm. *

The set X:= X u Xb is the generalized state set of the machine Y. The generalized stable


transition function is s : XxA P(X).

According to definition of the burst of a string, the last character of a burst is the output

value of the system for the corresponding state. Consequently, all states in a burst equivalent set

have the same output value. This implies that the following is true.

Lemma 2-14 The output value of a burst state x is the output value of any state in the

corresponding burst equivalent set S(x). +

Definition 2-15 Let Y = (A, Y, X, x0, f, h) be an asynchronous machine with a generalized


state set X = Xb u X, where X is the regular state set of Y and Xb is the burst state set of the

machine Y. Let x be a generalized state of the machine Y. The generalized output function h:


XxA Y of Y is defined as follows:

1) For all states x e X, set hg(x) = h(x);










2) For all burst state x e Xb, let S(x) be the burst equivalent set that is associated with the

burst state x. Set

hg(x):= h(x'), x' e S(x). *

For example, consider the machine X with transition table of Table 2-1, which has one

critical race s(x', c) = {x2, x3}. Using Algorithm 2-13, we can get the burst state set Xb = {x5}

and x5 represents the subset {x2, x3}. The generalized stable recursion function sg and the

generalized output function hg of the machine X can also be defined (Table 2-3).



Table 2-3. Stable transition table of the generalized X
a b c Y
x' x' x4 x5 0
X2 x' x4 X2 1
x x x x 1
X3 x X3 1
X4 x' X4 2
x x' X5 1



Definition 2-16 Let Y = (A, Y, X, x0, f, h) be an asynchronous machine with the stable


state machine = (A, Y, X, xo, s, h). Then, g = (A, Y, X, xo, sg, hg) is the generalized


machine associated with Y, where X is the generalized state set, s, is the generalized stable

recursion function, and hg is the generalized output function of the machine Y. +

When an asynchronous machine is enhanced into a generalized machine, it still keeps some

properties. We address two properties of the generalized machine in these two statements.

Lemma 2-17 Given an asynchronous machine Y with a state set X, which contains finite

number of states. Then the associated generalized machine Yg also has a generalized state set


X with finite number of states.









Proof Let us suppose the machine X has a state set X = {x1, x2, ..., x"} and the


generalized machine Y- has a generalized state set X = {x1, x2, ..., xn+t} From the definition

of burst equivalent set, any burst equivalent set S(3) is a subset of the state set X. Then, the

maximum number of burst equivalent sets cannot be larger than the number of subsets of X, i.e.,

2n, and hence is finite. +

Lemma 2-18 If the machine Y has no infinite cycles, neither does the machine Xg.

Proof Assume the machine Y has no infinite cycles but the generalized machine Xg

which is derived from the machine Y, has one infinite cycle x of length i, where i > 1. Suppose

that i generalized states x1, x2, ..., xi are involved in this infinite cycle x. The states x1, x2,...,

xi may be regular states or burst states. Let us consider the following two cases: i) If all these i

states are regular states. Then it implies that the machine Y has at least one infinite cycle x'.

And the infinite cycle involves the i states x, x2, ..., xi of the machine Y. It conflicts with the

assumption that the machine Y has no infinite cycles. ii) If in the i states x, x2, ..., xi there is

at least one burst state xp, where 1 < p < i. Suppose that the underling regular states of the burst

state x are x1, ..., x Then the infinite cycle x actually involves the following regular states:

1, ..., Xl, x xp+1 ..., xi, where 1 < j < k. Hence there is an infinite cycle x" that involves i

regular states of the machine Y. It conflicts with the assumption that the machine X has no

infinite cycles. This completes the proof. +

2.3 Observer

As depicted in Figure 1-1, we build an output feedback loop with a controller C, which is

also an asynchronous machine. Specifically, this controller C is composed of two asynchronous

machine: an observer B and a state-feedback control unit F (Figure 2-3).










Controller C


SI I!


x B



Figure 2-3. Control configuration for the closed-loop system YC

Here, the observer B estimates the uncertainty caused by critical races with the

input/output information of Y and generates estimate state of Y to feed F. With the external

input of the whole system and the estimate state of Y, the control unit F generates a sequence of

input to drive Y to match the model. We denote the controller C with (F, B).

We use the observer in a way that is similar as it is used in other branches of control

theory. Specifically, the observer here is an asynchronous input/state machine, which has two

functions: a) check if the asynchronous machine Y has reached its next stable state; b) use the

input/output information to estimate the current state of Y.


Let g = (A, Y, X, x0, s9, hg) be the generalized machine derived from Y. Similarly as in

Geng and Hammer (2005), we can build an observer that reproduces all stable and detectable

transitions of the machine Yg. The observer for g is an input/state machine B = (A x Y*, X,

Z, zo, c, I) with two inputs: the input character u e A of Y and the output burst 3 e Y* of


Y ; the state set Z is identical to the generalized state set X, and the initial condition is

identical to that of g i.e., z0 =x0. The recursion function o : Z x A x Y* Z of B is









constructed as follows. First, using the generalized stable recursion function sg, define the

function X: Z x Ax {0, 1} ->Z by setting

s(z, u) if a = 1;
X(z, u, a) := if a = (2-2)

Now, assume that the machine Eg is in a stable combination (x, ui1), when the input

character changes to ui, where (x, ui) is also a detectable pair. The change of the input character

may give rise to a chain of transitions of Yg. Let k > i be a step during this chain of transitions,

let 3k be the burst of Eg from step i to step k, and let uk be the input character of Eg at step

k. Since fundamental mode operation requires that the input character be kept constant during a

chain of transitions, we have uk =ui. Define

{s(x, uk) ifPk = 3(x, Uk);
Y(X, Uk, P S(U) Ik) U)= (2-3)
Skx I otherwise.

Let zk be the state of the observer B at the step k, while cok be the output of B. The

observer B is then an input/state machine defined by the recursion

Zk+1 a(Zk, uk, k)
B := (2-4)
Cok =Zk

The observer B is a stable state machine.

To describe the operation of the observer, assume that the observer switched to the

generalized state x immediately after Eg has reached the stable combination (x, ui-). Let p >

i be the step at which the chain of transitions from (x, ui) to the next stable state x' = sg(x, ui)

terminates; then, jp = 3(x, ui). As the pair (x, ui) is detectable, it follows by the definition of

c that the output of the observer B switches to the state x' at the step p+l.









We can now summarize the implications of our recent observations on the control

configuration Figure 1-1. Fundamental mode operation requires the output of the controller C to

remain constant while the system Eg is in transition. In order to fulfill this requirement, it must

be possible for the controller C to detect the point at which g has completed its transition

process. As discussed above, the output of the observer B switches to the state that represents

the next generalized stable state of Eg immediately after E has reached that state; this signifies

the end of the transition process and indicates the most recent stable state of Eg. In this way, the

observer B helps create an environment in which the machine Eg can be controlled in

fundamental mode operation.









CHAPTER 3
REACHABILITY OF A GENERALIZED MACHINE

The occurrence of critical races in an input/output asynchronous machine causes the lacks

of information about the exact state of the machine. We use the concept of generalized states to

deal with this uncertainty and keep a machine operate in fundamental mode. In this chapter, we

use generalized states to characterize the reachability properties of an asynchronous machine

with critical races.

First, let me introduce some important concepts that will be used in latter part of this

chapter.

3.1 Generalized Reachability Matrix

Definition 3-1 Let X = (A, Y, X, x0, f, h) be an asynchronous machine with the state set


X = {x1, ..., xn} and let g = (A, Y, X, x0, s9, hg) be the generalized machine associated with Y,


where X = { x1, ..., n, xn+1, ..., xn+t} is the generalized state set. The generalized one-step

reachability matrix R(g ) is defined as a (n+t)x(n+t) matrix with entry Rij, where Ri. is the

set of all characters a e A for which xi e sg(x', a) and for which the transition x' xi is a

detectable transition. If there is no such character a, then set Rij := N, where N is a character

not in the alphabet A. *

Note that when the generalized machine g is equal to the machine Y (i.e., when there

are no burst states), then the generalized one-step reachability matrix reduces to the one-step

reachability matrix R(X).

In view of the earlier discussion in Geng and Hammer (2005), only transitions that are

both stable and detectable can be used when constructing a controller. The stability of the

transition is guarantied by the generalized stable recursion function of the controlled machine Y.









However, the detectability of each transition needs to be checked according to proposition 2-10

in the construction of the generalized one-step reachability matrix. Therefore, each entry of the

generalized one-step reachability matrix characterizes if the machine Yg can go from one

generalized state to another through a stable and detectable transition.

Let Y = (A, Y, X, x0, f, h) be an asynchronous machine with the state set X = {x1, ..., x"}


and let g = (A, Y, X, x0, sg, hg) be the generalized machine associated with Y, where X = { x1,


..., x", x+, ..., xn+t} is the generalized state set and Xb = { xn+, ..., x+t } c X is the burst state

set. According to Definition 3-1, we can obtain the one-step reachability matrix R(X) of the

machine Y. For the generalized machine -g, the construction the generalized one-step

reachability matrix R( g) contains two tasks: 1) Determine the necessary burst states; and 2)

Add to the reachability matrix rows and columns corresponding to the burst states. Then, we can

divide R(Xg) into 4 blocks

[FR,,_IR1
ER11 12
R(Xg) = R |R '2
LR21 R22J

where, R,1 is nxn; R12 is nxt, R21 is txn, and R22 is txt. The matrix R,1 describes

one-step deterministic transitions among regular states of Y, while R22 describes one-step

transitions among burst states. The submatrix R12 represents one-step transitions from regular

states to burst states, while R21 represents one-step transitions from burst states to regular states.

Example 3-2 Consider the machine Y with transition table of Table 2-1, which has the

input alphabet A = {a, b, c}, the output alphabet Y = {0, 1, 2}, and the state set X = {x1, x2, x3,

x4, 5)}. There is a critical race pair (x1, c) in the machine Y.












The generalized machine Yg derived from the X has a generalized state set X= {x1, ..., x },


and it can be depicted as follows.


Table 3-1. Stable state transition table of the machine Yg

a b c Y
x' x1 x4 x5 0
X2 X1 X4 X2 1
x x3 xI x 1
X3 x X3 1
X4 x' X4 2
5 x X5 1



a


xl

b

a


b x4 X5 C



a a




x2 X3



Figure 3-1. State flow diagram of the machine g


According to Definition 3-1, the one-step reachability matrix of the original machine X is


a c c b


a c N b
R(X) =
a N c N


Sa N N b









and the generalized one-step reachability matrix of the machine Y- is

a N N b c

a c N b N


R(Yg)= a N c N N

a N N b N

a N N N c


The submatrix R22 = [c] in the matrix R(Yg) describes the stable and detectable

transitions inside the burst state set Xb. *

After obtaining the generalized stable recursion function s, we can get the generalized

one-step reachability matrix R(Yg) as above. Similar to the one-step transition matrix in

Venkatraman and Hammer (2004), we define some operations on the generalized one-step

reachability matrix R(Yg). Based on these operations we can obtain the overall view of reachable

states in the generalized machine Y and the information about how to approach the destination

of any transition. The latter information is very useful in the construction of the controller of the

closed loop system.

Definition 3-3 Let A* be the set of all strings of characters of the alphabet A and let wi

be a subset of A* or the character N, i = 1, 2. The operation U of unison is defined by

1 U w2 if w2 e A*
1W w if w1 e A* and w2 =N
wl w ., 2 (3-1)
W2 if w1 =N and w2 A
N if w1 = w2 = N.









The unison C := A U B of two nxn matrices A and B, whose entries are either subsets

of A* or the character N, is defined entrywise by Ci := Aij U Bij, i, j = 1, ..., n.

Note that N takes the role of zero.

abN NbN
For example, given two 3x3 matrices A= bNN and B= cN the unison of A
a a cabb

and B is

a b N
C :=AUB = b,c} N N
a {a,b} {b,c}

Definition 3-4 Let A* be the set of all strings of characters of the alphabet A and let wl,

w2 be two subsets of A* or the character N. Concatenation of elements wl, w2 e A* UN is

defined by

{w2w1 ifw1, w2 e A*
conc(w,w2) := if w w2 (3-2)
SN if w = N or w2 = N.

Let W = {w, w2, ..., wq} and V = {v, v2, ..., Vr} be two subsets, whose elements are

either subsets of A* or the character N. Define

conc(W, V) := U i = 1, ..., q conc(wi,vj) (3-3)
j = 1, ...,r

For instance, consider two subsets W = {a,N,{b,c}} and V= {N,a,{a,b}}. The

concatenation of W and V is

conc(W, V) = {a,aa,{aa,ba},N,{a,b},{b,c},{ab,ac},{ab,bb,ac,bc }.

Definition 3-5 Let C and D be two nxn matrices whose entries are either subsets of A*

or the character N. Let Cij and Dij be the (ij) entries of the corresponding matrices. Then, the

product Z := CD is an nxn matrix, whose (ij) entries Zij is given by









Zij := =1 conc(Cik,Dkj), ij = 1, ..., n. (3-4)

abN NbNJ
For example, consider two 3x3 matrices A = b NN and B= c NN .Then the
aac abb

product of A and B is

cb ba N
Z =AB= N bb N
{ac,ca} {ba,bc} bc

Using the operation of product, we can define powers of the generalized one-step

reachability matrix by setting

Rq(Xg) := R-'(g)R(g), q = 2, 3, ... (3-5)

Proposition 3-6 All transitions of the matrix Rq( g) are stable and detectable transitions.

Proof According to the definition of the generalized one-step reachability matrix, each

non-zero entry of the matrix refers to a stable and detectable transition. After the operation of

product, every non-zero entry of a generalized multi-step reachability matrix refers to a

combination of multi-step stable and detectable transitions. This operation does not change either

the stability or the detectability of the transitions. Thus, for every entry of the matrix R'q(g), if it

is not N, it stands for a stable and detectable transition. *

Based on Proposition 3-6, for an integer q > 1, the matrix Rq(Xg) describe if the machine

can reach one state from another state through exact q stable and detectable transitions. If the

(ij) entry of the matrix Rq(Xg) is not N, then it is the set of all input strings that can takes the

machine Yg from the state x' to the state xj via a q-step stable and detectable transition. If the

(ij) entry is N, then it is impossible to reach the state xj from the state x' in exact q stable and

detectable transitions. Though, it might be possible to reach from x' to xj in 0 stable and









detectable transitions, where 0 < q or 0 > q. Since we are also interested in if the machine can

reach one state from another in fewer transitions, it is needed to construct a multi-step

reachability matrix.

Definition 3-7 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine and

let R(Xg) be the generalized one-step reachability matrix of the machine Xg. The generalized q-

step reachability matrix is defined by

R(q( g) := r= 1.... qRr(Xg), q = 2, 3, ... (3-6)

Note that the (ij) entry of R(q)(Xg) contains the reachability information from the state xi

to the state xj. If the (i,j) entry is not N, it consists all strings that may take the machine Yg

from x' xJ through stable and detectable transitions in q or fewer steps. It leads to the

following statement and its proof is similar to Murphy, Geng and Hammer (2003).

Lemma 3-8 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine with n

states and t burst states, and let R(Xg) be the generalized reachability matrix of the machine

Yg. Then the following two statements are equivalent:

(i) The generalized state xJ is stably reachable through a detectable transition from the

generalized state x'.

(ii) The (ij) entry of R(n+t 1)(yg) is not N. *

Proof Let A* be the set of all strings of characters of the alphabet A.

If the first statement is true, namely, the state xj is stably reachable from x', then there is

an input string u' := uk-...ulu0, which satisfies xj = sg(x, u) and the transition from x' to xj is









detectable. Here, u' e A* and k := lu'|. If lu'l < (n + t 1), then take u := u'. Thus, the (i,j)

entry of R(n+t-1)( g) is Uk-l...U1U0.

If lu'l > (n + t 1), then we need to show that a shorter string u* in the string u' still

satisfies x = s (x', u*) and lu*| < (n + t 1). Define recursively a string of states x0, x1, ..., Xk,

by setting x0 := x' and Xm+1 := s,(xm, um), for m = 0, 1, ..., k-1. This implies xk = x. The length

of the string x0, ..., Xk, k +1 is greater or equal to (n + t). However, there are totally only (n + t)

distinct generalized states of the machine Yg. So, at least one generalized state must be repeated

in the string x0, ..., xk. Suppose xp = xq, for 0 < p < q < k. Remove from u' the string v, which

satisfies x' = s (xP, v). Afterwards the shortened input string

u" := uoul.. .UpUq...uk1 Uk... UqUp-l...U U0 (or U" := Uk-l.. .Uq when p = 0).

This shortened u" still satisfies xJ = s (x', u"). Keep shortening the input string until an

input string u* of length |u*| < (n + t 1) still satisfying x = s (x', u*).

Conversely, if the second statement is true, then it implies that there is an input string u e

A* of length ul < (n + t 1). Suppose the (i,j) entry of R(n+t 1)('g) is the string

u := uk-...U1U0. There are k input characters in the string u and those characters satisfy the

following equation:

setting x0 := x' and xm+1:= s,(xm, um), for m = 0, 1, ..., k-1. Then we have xk = x.

According to the definitions of the generalized stable recursion function, the machine Y is only

involved into stable transitions here. Meanwhile, in the construction of the generalized

reachability matrix R(Xg), all undetectable transitions are eliminated. Thus, the generalized (n + t









- 1) step reachability matrix R(n+t 1)(g) only contains detectable transitions. So, the generalized

state xJ is stably reachable from the generalized state x'. *

Therefore, all possible stable and detectable transitions for the machine Yg can be found

in the matrix R(n+t1)(Xg), i.e., the generalized (n + t 1) step reachability matrix characterizes

the reachability property of the machine Yg with n states and t burst states.

Definition 3-9 Let R(Xg) be the generalized one-step reachability matrix of the machine

Xg, which has n + t generalized states. The generalized stable reachability matrix of the machine

Eg is F( g):= R(n+t 1(g).

Example 3-10 Consider the machine X and the generalized machine Yg of Example 3-2.

X has a generalized state set X = {x x2, 3, x4, x5} and (n +t 1)= 4.

Raise the power of the R(Xg) as follows:

{aa,ba,ab,bb,ac} N N ac {ca,cb,cc} -

ba {cc,ca} N {ac,aa} N

R2() = {aabaac} N cc N ca

{ab,bb,ba} {cc,ca} N {ac,aa} cb

S {aa,ba,ac} ca N {aa,ac} {ca,cc} -

After a stable transition, repeat applying the same input character will not change the state

of the machine. Thus, all same consecutive input character can be replaced by one character. For

instance, the input string "aa" can be replaced by "a" and it will not affect the stable transitions

of the machine. Hence, we obtain









{a,b,ba,ab,ac}

ba

{a,ba,ac}

{b,ab,ba}

- {a,ba,ac}


N

{c,ca}

N

{c,ca}

ca


Continue raising the power of the R(Xg) until the (n + t


ac

{a,ac}

N

{a,ac}

{a,ac}


{c, ca,cb}

N

ca

cb

{c,ca}


1) = 4. Then we have


- j a,ab,ac,aba,aca 1
[abab,abac,acab,acacJ

fa,ab,ac,aba,abc1
Saca,abab,abac J

fa,ac,aba,aca1
Sabac,acac J

{a,ab,aba,aca{
a abab,acab

f a,ac,aba,aca{
- 1 abac,acac


Nb,ba,bab,bac1
N N 1 baba,baca J

N b,ba,bc,bab,bac1
c N lbaba,babc,baca


N c {ba,bc,bac,baca,baba}


N N {b,ba,bab,baba,baca}


N N {ba,bac,baba,baca}


fc,ca,cab,cac1
Scaba,caca J

Sca,cab,cac,cabal
1 cabc,caca J

{ca,cac,caba,caca}


{ca,cab,caba,caca}


{ c,ca,cac,caba,caca}


According to Definition 3-7 and Definition 3-9, we obtain the generalized stable

reachability matrix F(Eg) of the machine g as follow:


R2(yg)


R4 ()









f a,ab,ac,aba,aca N Nb,ba,bab,bac c,ca,cab,cac
abab,abac,acab,acacJ baba,baca J caba,caca

fa,ab,ac,aba,abc fNb,ba,bc,bab,bac1 Ica,cab,cac,cabal
aca,abab,abac c N Ibaba,babc,bacaJ cabc,caca J

fa,ac,aba,aca
F(Yg)= aaba,acac N c {ba,bc,bac,baca,baba} {ca,cac,caba,caca} .*
abac acac

a,ab,aba,aca N N {b,ba,bab,baba,baca} {ca,cab,caba,caca}
Saabab,acab

Sacabaaca N N {ba,bac,baba,baca} { c,ca,cac,caba,caca}
L abac,acac 1 -

3.2 Common-output Generalized States

As we have mentioned before, if the outcomes of a critical race (r, v) can be divided into

more than one burst equivalent set, then sg(r, v) consists of more than one generalized state. This

situation is shown in the generalized one-step reachability matrix as one input string appears

more than once in different entries of a single row. Consider both this fact and the Lemma 3-8,

we have the statement below.

Proposition 3-11 Let g = (A, Y, X, x0, s9, hg) be a generalized asynchronous machine

with n states and t burst states, and let F (Eg) be the generalized stable reachability matrix of

the machine Eg. Then the following two statements are equivalent for all input strings u E A

and for all j = 1, ..., n + t.

(i) Applying u at the generalized state x' generates a critical race.

(ii) The string u appears in more than one entry of row i of the matrix F (E ). *

Note that the above conclusion is similar with the Proposition 4-16 in Venkatraman and

Hammer (2004).









After using burst states to represent subsets of states which have the same output value and

same burst, we can see there are still critical races in the machine on the generalized state base.

Some input strings may be repeated in more than one entries of a row of the generalized stable

reachability matrix (Example 3-10). That is caused by the existence of critical races.

In the present discussion, the machine F(Xg) is an input/output machine. Thus, what

matters to the user are the output value but not the state of the machine itself. Next we check if

the machine can be led from different outcomes of the critical races to the same output value. In

Venkatraman and Hammer (2004), if the machine can be led from different outcomes of the

critical races to the same state, then it means the existence of a feedback trajectory. Here, we can

loose the restriction to a subset of states which have the same output value. They can be also

called "common-output generalized states".

Definition 3-12 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine with


a generalized state set X = Xb u X. Assume that the machine g have p critical races (rl, v1),

(r2, v2), ..., (rp, vp), and let T(ri, Vi) := {r,, ,..., rmii)} c X be the set of all outcomes of the

critical race (ri, vi), i =1,..., p. Divide T(ri, vi) into subsets CI, C C, ..., Cmi) according to the

output value of ri, ri, ..., rm"i). These subsets C!, C;, ..., C"mi) can be represented by x1, x2,...,

xo, which are called the common-output states of the machine Yg. Set m = n + t + c, then the


generalized state set increases to X = {x1, ..., xm}.

Note that the outcomes of the critical races may be a combination of burst states and

regular states. Moreover, if two subsets contain the same states, then they are represented by the

same common-output state.










After introducing the common-output state of the machine -g, we should update the

generalized one-step reachability matrix R(Xg) and the generalized reachability matrix F(Xg).

The transitions from one generalized state to a subset of states, which have the same output

value, will be replaced by the single transition from the starting state to a newly defined

common-output state.

Example 3-13 Consider a machine Y with the input alphabet A= {a, b, c}, the output

alphabet Y = {0, 1, 2}, and the state set X = {x x2, 3, x4, x5}. There is a critical race pair (x1,

c) in the machine Y (Table 3-2).

Table 3-2. Transition table of the machine _
a b c Y
x' x' X4 {x4, 5} 0
X2 X2 X1 X2 1
x2 x2 xI x 1
x3 x X5 X3 1
x4 x' X4 X2 2
x5 x2 x5 X3 2



The stable state machine of Y is s, (Table 3-3).

Table 3-3. Transition table of the machine I
a b c Y
x1 x1 X4 {x2, X3} 0
x2 X2 X4 X2
x3 x X5 X3 1
x4 x' X4 X2 2
x5 x2 x5 X3 2



Using Algorithm 2-13, we get the generalized stable recursion function sg of the machine

Yg. Associate a burst state x6 with the subset {x2, x3}. Then the generalized machine g has a


generalized state set X = {x1, ..., x6}. Assign a common-output state x7 to represent the subset










{x4, x5}. Now the generalized machine g has a generalized state set X:

7. The generalized one-step reachability matrix is

-a N N b N c


R(Xg)


N {a,c} N


a N


: {1, ..., Xm} and m


b N N


c N b N N


a c N b N N N


N a c

a a N


La

Definition 3-14 Let


N N

c b


N N N c b J

s be a generalized stable state machine with the generalized stable


recursion function sg, and let u be an input string of Xs The transition induced by u from a

generalized state x is a deterministic transition if sg(x, u) consists of a single state. The

machine Yg is a deterministic machine if all transitions of ,s are deterministic. *

3.3 Output Feedback Trajectory

From the definition of the critical race, we know a transition from a critical race pair to the

outcomes is not a deterministic transition. So, originally, an asynchronous machine with critical

races is not a deterministic machine. If we can transform the machine with critical races into a

deterministic machine on a specific basis, then we actually get rid of the effect of the critical

races to the machine. The following procedure helps us to transform a machine with critical races

into a deterministic machine.

Consider a generalized machine Yg with the generalized state set {x1, x2, ..., xm} and

input set A. Assume the machine has a critical race (xJ, v) with the outcomes {xp, xq} and p









q. If the h(xP) = h(x'), then we can define a burst state or a common-output state to make the

transition from xJ to the subset {xp, x'} a deterministic transition. So, we only need to focus on

the situation that the outcomes have different output values, i.e., h(xP) h(x'). In another word,

we cannot make the generalized machine Yg transform into a deterministic one simply with a

generalized state set. When h(xP) h(x'), if there exist input strings which can take the machine

Y from these two states (also two output values) to a single generalized state xs through

deterministic transitions, then the effect of the critical race (xi, v) can be eliminated. Assume

there exist input strings u1, u2 E A, where u1 takes the machine from xp to xs

deterministically and u2 takes the machine from x' to xs deterministically. That means s (x',

u1) = s g(x, u2) = Xs. Then we can generate a deterministic transition from xj to xs with

introducing an output feedback controller to the machine g as follows: After applying the input

v at the state xj, check the outcomes. If the outcome is xp, then apply u1 to the machine. If the

outcome is x', then apply u2 to the machine. Hence, based on the generalized state set of a

generalized machine -, we can turn the machine Y into a deterministic machine with an

output feedback controller. This controller sets up a standard projection on the generalized state

set X, which can be denoted by I : X x A X : Hx(x, u) = x.


Definition 3-15 Let g be an asynchronous machine with the generalized state set X =

{x1, ..., xm}, the input alphabet A, and the generalized stable recursion function sg. An output

feedback trajectory from the generalized state xJ to the common-output state x' is a list {So, S1,

..., S } of sets of valid pairs of Ys with the following properties:

(i) s, (x, u) is a detectable transition for all (x,u) e Uj ,..,p Sj,









(ii) S = {(x, uo)},

(iii) sg [S J c x[SJ+], a = 0, ..., p-l,

(iv) s, [S] = {x'}. *

For example, consider the machine g in Example 3-13. The output feedback trajectory

from the generalized state x6 to the generalized state x4 is: So = {(x6, a)}, S, ={(x1, b), (x2, b)}.

Proposition 3-16 Let g be a generalized machine and let xj and x' be two generalized

states of Yg. The following two statements are equivalent.

(a) There exists an output feedback controller C that drives Yg through a deterministic

transition from xJ to x'.

(b) There is an output feedback trajectory from xj to x'.

Proof Suppose that the part (b) is valid, and let { So, S,, ..., S } be an output feedback

trajectory from xj to x. We construct an output feedback controller C that takes the machine

g from xj to x' through a string of deterministic transitions. This output feedback controller C

has two inputs: one is the output burst 3 e Y* of -g, and the other is the external input veA,

which is also the command input of the controller C. Given a set of characters WEA, we need

to construct a controller C(xj, x', W). This controller takes the machine g from generalized

state xj to x' through deterministic transitions as the response to an input character weW.

The controller C(xj, x', W) is a combination of an observer B and a control unit F as

shown in Figure 2-3. Where, the observer is an input/state machine B = (A x Y*, X, Z, z0, C, I)

with two inputs: the input character u e A of Yg and the output burst 3 e Y* of Yg; the state


set Z is identical to the generalized state set X, and the initial condition is identical to that of









Yg, i.e., z0 =x0. And the function o : Z x A x Y* Z is the stable recursion function of B. Let

zk be the state of the observer B at the step k, while cok be the output of B. The observer B

is an input/state machine defined by the recursion (Eq. 2-4)

The control unit F is also an input/state asynchronous machine F = (A x X, A, E, ,0, 4,


r) with two inputs: the external input veA and the output co eX of the observer B. To

complete the construction of the controller C, we need to derive the recursion function ) and

the output function r of the unit F.

According to Figure 2-3, as long as vVW, the controller C stays in its initial state (zo, o0)

and the input character u of the machine Yg equals to the external input character v. After the

machine g arrives a stable combination of state xJ, if the external input v changes to a

character of W, then the controller C starts working. The observer B collects both the input

character u and the output burst 3(xj) of the machine Y- and feeds the control unit F with the

state of the machine The control unit F generates a string of characters u1u2...ur and apply

it to the machine Yg. This input string u1u2...ur will drive the machine Y from the state xi to

x' through a string of detectable and stable deterministic transitions.

Recalling that the control unit is an input/state asynchronous machine F = (A x X, A, E,


40), r). The recursion function of F is a function ):ExXxA-> E and the output function of F


is denoted by r : ExXxA-> A. Referring to Figure 2-3, the output coeX of the observer B is

one of the inputs of F, and the other input is the external input veW. Then the control unit F

generate the string ueA to feed the controlled machine Yg. Note that the control unit F must









operates in a fundamental mode, so the whole system must have reached a stable combination

before the F generates the next input character for Yg. Assume that F will generate r input

characters u u2...ur to feed Zg, then it needs r states l1(xj, w), ,2(xj, w), ..., r(xJ w). Denote

this set by E(x, w):={,(xj, w), 2(x, w), ..., r(x, w)}. We define the recursion function 4 and

output function r of the F as follows.

(i) Let U(xJ)cA be the set of all input characters that form stable combination with the

generalized state xJ, and let z0 be the initial state of B and o0 be the initial state of F. Set


W(o, (z, t)) := o for all (z, t)e XxA\xJU(xJ),

W(,o, (x, v)) := 1(XJ) for all veU(xJ).

Where ,1(xj) is the state of F, when observer B detects a stable combination with xj.

When both B and F are at initial states, the controller C(xJ, x', W) directly applies the

external input v to -g, thus set


r&(S, (z, v)) :=v for all (z, v)e XxA.

(ii) When the observer B detects a stable combination of g with the generalized state

xj, suppose the external input switches to a character weW. We choose a character uj U(xJ)

and set

rJ(( x'), (x, t)) := uj for all tU(xJ).

In this way, the machine Yg lingers in the state xj when the external input switches to a

character of W. Hence, the fundamental mode operation of the machine is guaranteed. Then the

control unit F will generate an input string u=u u2...ur to drive the machine g to the








generalized state x'. Since we have a output feedback trajectory {So, S, ..., Sp}, we need P new

states for F, where

P = #nSo + #nS + ...+ #ISp. (3-7)

Denote the state of F as ,k(xJ, w, x), where xelHSk and k = 1, ..., p. When the input

character switches to w, the control unit F moves to the state $o(xJ, w, xJ) and it begins to

generate the first input character u0 to feed g,, where uOeA is an character that satisfies (xj,

u0)E So. To implement this, we set

(,(xJ), x', w) := (xJ, w, xj) for all w e W;

(,(xJ), xj, v) := (x ) for all v e U(x) \ W;

(,(xJ), x, v) := 0 for all v V U(x) U W;

r((xj, w, x), x, v) := u0 for all (x, v) e XxA.

After u0 is applied to Xg the machine will move to a generalized stable combination with

a generalized state x IexS After the observer B detects this transition, the control unit F

moves to the next state (l(xj, w, x) and generates the next input character u1 to the machine

Y. The process continues similarly until x' is reached. Then at the step ke { 1, 2,..., p} the

function 4 and r must be defined as follows

(k-l(xJ, W, Xk-1), Xk, W) := k(XJ, Xk),

r(k(xj, Xk), ) := Uk for all (x, v)eXxA.

At the k=p step, the machine Yg reaches a state x pExS,. Set s (xp, up) := x', where u

is an character satisfies (x, up)eSp. This is accomplished by setting

(WP(x w, xi), Xk, W) := P+1(xw, W, ),









(~P+((xJ, w, xi), z, t) := 0o for all (z, t)eXxA\W.

r(P 1(x, x, x, v) := Up for all (x, v)eXxA.

As long as the external input remains as a character in set W, the machine will linger in

the stable combination (xl, up). If the external input is no longer belongs to W, the controller

returns to the initial state o,. We build the controller as well as prove that statement (b) implies

statement (a) as above.

Conversely, assume part (a) is valid. Let o be the initial state of the controller C and let

C(,, x, u) be the output value produced by the controller C when it is at the next stable state

corresponding to its state ,, Yg is at the state x and the external input is u. By assumption,

there is an external input value w that induces the controller C to generate an input string

Si
u0u1...u, for the machine Xg from the generalized state x to the generalized state x via

deterministic transition. The first character of this input string is u0 = C(o0, x, u). Define the set

So := {(x, uo)}

Let sg be the generalized stable recursion function of Yg. when the input from the

controller to Y, changes to u0, Yg moves to a generalized stable combination with one of the

states of the set s(x, u0) = s[S]. When this state is reached by Y the controller C detects the

new state and controller moves to its own next stable state. Let (x, u0) be the next stable state

of the controller and let u1 = C((x, u0), x, w) e A be the output character generated by the

controller once C reaches (x, uo). Define the set

Si := {(x, C(t(x, uo), x, w)) :x e s(So)}.

Continue operating like that until the set Sk, k > 0, is defined. Build a new set by setting









Sk+1 := {(x', C((x', Uk), x', W)) : X' S(Sk)}


By assumption, the controller C drives Y to the state x through deterministic transitions.
i
Consequently, there exists an integer p such that s(Sp) = x. Then, the list So, S,, ..., S, forms a

output feedback trajectory. So, the existence of a output feedback controller C that drive Y

j i
from x to x through deterministic transitions, implies the existence of a output feedback
j i
trajectory So, S, ..., SP from x to x. Namely, part (a) implies part (b). This completes the

proof. +

3.4 Preliminary Generalized Skeleton Matrix

Using the algorithm in the above proof of Proposition 3-16, we can check the basic

connection between any two generalized states of the machine g, namely, the existence of the

output feedback trajectory between any pair of generalized states. If we focus on the stable and

detectable reachability properties, then we don't need to record all the input strings. Instead, we

can use a numerical matrix, which has only entries of one and zero, to represent this one-step

stable and detectable reachability properties. This numerical matrix can be called "preliminary

generalized skeleton matrix" of the machine Yg. Afterwards, it is easier to calculate the power of

it and obtain the "overall preliminary generalized skeleton matrix". For the machine Y with m

generalized states, the overall preliminary generalized skeleton matrix characters all the stable

and detectable transitions among the generalized states of the machine within m steps. We can

use the following algorithm to gradually transform the generalized one-step reachability matrix

into the preliminary one-step generalized skeleton matrix. Meanwhile, the machine g is

transformed into a deterministic machine on the generalized state basis with an output feedback

controller C.









An operation involving strings of A and zero and 1 should be defined before giving the

algorithm.

Definition 3-17 Let o be a character not included in A. The meet operation between two

strings of A+ and zero and 1 is defined as follow:

0 A := 0, 0 A 1 = 1 0 := 0, 1 1 := 1,

0Aa= a A 0:=0, 1a=a Al :=o, forall aeA+.

The meet of two vectors with r > 1 components is defined entrywise as the vector of the

meets of the corresponding components. *


Algorithm 3-18 Let Yg be an asynchronous machine with the generalized state set X =

{x1, ..., xm} and let F( g) be the generalized stable reachability matrix of the generalized

machine Yg.

Step 1: Transpose the matrix F(Xg) and denote the resulting matrix by F'(Xg).

Step 2: Replace all entries of N in the matrix F'(Xg) by the number 0; denote the

resulting matrix by K1.

Step 3: Perform (a) below for each i,j = 1, ..., m; then continue to (b):

1
(a) If Ki includes a string of A+ that does not appear in any other entry of the same
1 1
column j, then replace entry Ki by the number 1. Otherwise, let the entry Ki remain.

(b) Set k:= 1 and denote the resulting matrix by K(k).

Step 4: If all entries of row k of the matrix K(k) are 1 or 0, then set K(k+l):= K(k)

and set k := k+1.

Step 5: If k = m + 1, then set K1,( ) := K(k) and terminate the algorithm. Otherwise, go

to step 6.









Step 6: Perform the following operations:

(a)If there is a character u e A that appears in row k of K(k), then let j1, ..., q be the

columns of row k of K(k) that include u. Denote by J(u) the meet of rows Ji, j,q of the

matrix K(k).

(b) If J(u) has no entries other than 0 or 1, then delete u from all entries of row k of

the matrix K(k); set all empty entries, if any, to the value 0. Continue to (c).

(c) If J(u) has no entries of 1, then return to Step 3. Otherwise, continue to (d).

(d) If J(u) has entries of 1, then let j1, ..., Jr be the entries of J(u) having the value 1.

Let S(k) be the set of rows of K(k) that consists of row k and of every row that has the

number 1 in row k. In the matrix K(k), perform the following operations on every row of

S(k):

Delete from the column all occurrences of input characters that appear in columns j j2,

.., jr of the row.

Replace rows j J2 ..., of the column by the number 1.

If any entries of K(k) remain empty, then replace them by the number 0. Return to Step

4.

The final resulting matrix K,(g ) is called the preliminary generalized skeleton matrix of

the machine g. +

Definition 3-19 The outcome Ki(Xg) of Algorithm 3.23 is defined as the preliminary

generalized skeleton matrix of the generalized asynchronous machine Y. +









Note that this preliminary generalized skeleton matrix Ki,(g) of the generalized machine

g is similar to the one-step fused skeleton matrix A(X) of an asynchronous machine Y in

Geng and Hammer (2005).

Example 3-20 Consider the machine Y and the generalized machine Y of Example 3-

10. The generalized stable reachability matrix F(Xg) of the machine g is



f a,ab,ac,aba,aca 1 fb,ba,bab,bac1 fc,ca,cab,cac1
labab,abac,acab,acac N N baba,baca J 1 caba,caca J

fa,ab,ac,aba,abc1 fb,ba,bc,bab,bac1 fca,cab,cac,cabal
aca,abab,abac c N Ibaba,babc,bacaJ cabc,caca J


F(Sg)= a,ac,aba,aca N c {ba,bc,bac,baca,baba} {ca,cac,caba,caca}
i abac,acac I

{a,ab,aba,aca1 T T (i ] f
aababa,acab N N {b,ba,bab,baba,baca} {ca,cab,caba,caca}
Saacabab,acab

S a,ac,aba,aca N N {ba,bac,baba,baca} { c,ca,cac,caba,caca}
abac,acac J



Applying the Algorithm 3-18 to the matrix F(Xg), we can obtain the preliminary

generalized skeleton matrix Kl,(g) of the machine g is

S1 0 0 1 1

1 1 0 1 1

K,(Xg) = 1 0 1 1 1 .

1 0 0 1 1

1 0 0 1 1 -









Proposition 3-21 Let Yg be a generalized machine with the preliminary generalized

skeleton matrix K1(Xg), and let x' and xj be two generalized states of Xg. Then the following

two statements are equivalent.

(a) There exists an output feedback trajectory from xi to x'.

(b) The (i, j) entry of Ki,(g) is 1.*









CHAPTER 4
MODEL MATCHING FOR INPUT/OUTPUT ASYNCHRONOUS MACHINES WITH
RACES

In the present chapter we start to address the model-matching problem for input/output

asynchronous machines with critical races. In last chapter the reachability properties of an

input/output asynchronous machine with critical races has been discussed and corresponding

generalized machine has been derived. The newly defined generalized machine with related

generalized state set and generalized functions of the machine could be controlled as a

deterministic machine without critical races. The control of this kind of asynchronous machines

has been discussed in Geng and Hammer (2005). Thus, the controller will be designed to correct

the input/output machine under the configuration of Fig. 2-3, so that the closed-loop system

possesses an equivalent input/output behavior as that of a prescribed model.

Since we are discussing the input/output machines, we first study the equivalent list of

the generalized machine Zg, with respect to the model 2'. Then we work on the sufficient and

necessary conditions of the existence of the output feedback controllers so as to solve the model

matching problem. When such a controller exists, we provide an algorithm to construct the

controller. Finally, an example is presented to illustrate how the control system operates.

4.1 Model Matching Problem

As we mentioned before, the design of a controller to eliminate the effects of critical races

of an existing asynchronous machine is called the Model-Matching Problem. Specifically, the

formal statement of the model matching problem is as follows. Let Y be a machine that exhibits

undesirable behavior. Assume that the desirable behavior is specified by an asynchronous

machine Y'. The machine 2' is called the model. Our objective is to design a controller C for

which the behavior of the closed loop system 2c simulates the behavior of the model Y'. It is









indicated in Kohavi (1970) that the practical performance of an asynchronous machine is

determined by its stable-state behavior. Thus, the stable-state behavior of YC need to be

equivalent to the stable-state behavior of 2'. Let us first introduce the classical notions of

equivalence.

Definition 4-1 Let Y = (A, Y, X, x0, f, h) and 2' = (A, Y, X', C0, f, h') be two machines

having the same input and the same output sets, and let Es and E's be the stable state machines

induced by and 2', respectively. Two states xe X and C e X' are stably equivalent (x )

if the following conditions are true: When Es starts from the state x and E's starts from the

state C, then (i) Es and 2's have the same permissible input strings; (ii) Es and 2's generate

the same output string for every permissible input string. The two machines and 2' are

stably equivalent if their initial states are stably equivalent, i.e., if x0 Co. *

Note that two machines Y = (A, Y, X, x0, f, h) and 2' = (A, Y, X', C0, f, h') that are stably

equivalent appear identical to a user.

Definition 4-2 Given a machine Y and 2', find necessary and sufficient conditions for the

existence of a controller C such that YC is stably equivalent to 2' and operates in fundamental

mode. If such a controller C exists, derive an algorithm for its design. *

In this dissertation, the model matching problem concentrates on matching the stable

input/output behavior of the model. The model 2' can be taken as a stably minimal machine.

Let Yg be a generalized machine with the generalized state set {x x2, ..., xm} which is induced

from the machine Y. Our objective is then to match the input/output behavior of the generalized

machine Y and the model Y'.









Next, let us introduce a notion which underlies the solution of the model matching problem

for asynchronous machines. Given two sets S1 and S2 and a function g: S1 -> S2, denote by

g' the inverse set function of g; i.e., for an element s e S2, the value g' (s) is the set of all

elements a E S1 that satisfies g(a) = s.

Definition 4-3 Let X = (A, Y, X, s, h) and X' = (A, Y, X', C0, f, h') be two machines


having the same input and the same output sets. Let g = (A, Y, X, s9, hg) be a generalized

machine induced from the machine Y. The state set X' of X' consist of the q state (C, ..., .

Define the subsets Ei := hglh'(C) c X, i = 1, ..., q. Then, E(Eg, Z') := E1, ..., E'} is the output

equivalence list of Yg with respect to X'. +

An equivalence list is characterized by the following property: the value of the output

function hg of Yg at any state of the set E' is equal to the value of the output function h' of X'

at the state ('. The members of an output equivalence list are not necessarily disjoint sets.

Definition 4-4 Let Yg be a generalized machine with generalized state set X = {x1, ...,


xm}, and let A1 and A2 be two nonempty subsets of X. The reachability indicator r(Xg, A1, A2)

is defined as 1 if every element of A1 can reach an element of A2 through a chain of stable

and detectable transitions; otherwise, r(Yg, A1, A2) = 0.


Example 4-5 Let Yg be a generalized machine with generalized state set X = {x1, x2, x3}

and the preliminary generalized skeleton matrix is K,(Yg),









1 1 1

Ki(Xg) = 1 1 1 .

0 0 1

Let A1 = {x1, x2} and A2 = {x2, x3} be two state subsets. Then

r(g, A1, A2)= 1.

Definition 4-6 Let Yg be a generalized machine with generalized state set X = {x1, ...,

xm}, and let A = {A1, ..., A'} be a list of m > 1 nonempty subsets of X. The fused skeleton

matrix A(Xg, A) of A is an qxq matrix whose (i,j) entry is

Aij(Xg, A) = r(Xg, A1, AJ).

Example 4-7 Consider the machine Y and the two state subsets A1 and A2 in the

Example 4-5. Let A := {A, A2} be a list of subsets of X. Then the fused skeleton matrix A(g,

A) of A is A(Xg, A).


A(Xg, A) 1


Definition 4-8 Let A = {A1, ..., A'} and W={W1, ..., W} be two lists of subsets of X.

The length of the list A is the number q of its members. The list W is a subordinate list of the

list A, denoted as W -< A, if it has the same length q as the list A and if W' A' for all i=1,

..., m. A list is deficient if it includes the empty set 0 as one of its members. *









4.2 Existence of Controllers

Next, we give the condition of the existence of a controller C for which 2c is stably

equivalent to a specified model 2'. Given two pxq numerical matrices A and B, the

expression A > B indicates that every entry of the matrix A is not less than the corresponding

entry of the matrix B, i.e., Ai > Bij for all i = 1,..., p and for all j = 1, ..., q.


Lemma 4-9 Let g=(A, Y, X, x0, s9, hg) and 2'=(A, Y, X', s', h') be asynchronous

machines, where 2' is stably minimal. Let X'= {f, ..., q}) be the state set of 2', where the

initial condition of 2' is C0 = Cd. Assume that there is a controller C for which 2c is stably

equivalent to 2' and operates in fundamental mode. Then, there is a non-deficient subordinate

list A of the output equivalence list E(Xg, 2') for which A(Xg, A) > K(2') and x0 e Ad. *

The proof of the above Lemma 4-9 is similar to the proof of the Lemma 4.11 in Geng and

Hammer (2005). The difference is that a generalized machine appears here instead of a regular

machine Y. Recall that all the underlying states of a burst state or a common output state in the

generalized machine g are the same states in the original machine Y and those underlying

states have the same output value. Thus, when applying a real input character u to the

generalized machine g at a generalized state x', it is the same to apply this input character u

to the real machine Y at any underlying state of that generalized state x'. The real machine Y

will generates the same output value as the generalized machine g does.

The condition of Lemma 4-9 is not only a necessary condition, but also a sufficient

condition for the existence of a controller to solve the model matching problem. The inequality

A(Xg, A) > K(2') guarantees that the corresponding output values of the two machines match. If

the model 2' has a stable transition from a state C to state C, then the machine Y has a









stable and detectable transition from every state in A' to a state in Aj. Thus, we only need to

construct a controller C which generates the input string that takes g from a state in A' to a

state in Aj. This controller should be a combination of an observer and a control unit as

described in Figure 2-3.

Theorem 4-10 Let -g=(A, Y, X, x0, sh hg) and 2'=(A, Y, X', s', h') be stably reachable

asynchronous machines, where 2' is stably minimal. Let X'= {~ 1., ...} be the state set of

2', where the initial condition of 2' is C0 = Cd. Then the following two statements are

equivalent.

(i) There is a controller C for which C, = ', where YC operates in fundamental mode

and is well posed.

(ii) There is a non-deficient subordinate list A of the output equivalence list E(2g, Y')

such that A(Yg, A) > K(2') and x0 e Ad.

Moreover, when (ii) holds, the controller C can be designed as a combination of an

observer B and a control unit F as depicted in Figure 2-3 and the observer is given by Equation

2-4.4


Proof The generalized machine Y has a generalized state set X = {x1, x2, ..., xt}. All the


underlying states of the generalized states {x1, x2, ..., xt} in this set X are the same states in the

set X of the original machine Y. Since all the real states included in a burst state or in a

common output state will work with the same input value, the same input value can be used on

the real machine. Furthermore, the output of a burst state or a common output state is the same as

that of the underlying states. Hence, the operation of the real machine is as same as before the









introducing of the generalized machine. Thus, we can use the same method in Geng and Hammer

(2005) on the generalized machine, i.e., to find a controller for Xg.

The Lemma 4-9 indicates that statement (i) implies statement (ii). Now let us assume that

(ii) is valid. Let A={A1, ..., A'} be a subordinate list of E(Xg, 2') satisfying A(Xg, A) > K(2')

and x0 e Ad. Using A, we build a controller C for which the closed loop system Xc of 1.1 is

stably equivalent to the model X', is well posed, and operates in fundamental mode. The

controller C is a combination of an observer B and a control unit F as depicted in Figure 2-3.

The observer B is given by Proposition 3-16, so we complete the proof by constructing the

control unit F. Recall that the control unit F is an asynchronous machine F = (A x X, A, E, o0,


q, r) with two inputs: the external input veA and the output co EX of the observer B. To

complete the construction of the controller C, we need to derive the recursion function 4 and

the output function r of the unit F.

Assume that X' is at the stable state C and that Y is at a stable state x e A'. Note that C

is either the initial condition x0EAd of 2' or the outcome of a detectable stable transition; x is

either the initial condition x0EAd of g or the outcome of a detectable stable transition.

Assume the external input character switches to the character w. Then the model X' moves to

its next stable state s'(C', w)=j. Recall that s9 is the generalized stable recursion function of Xg.

The inequality A(Xg, A) > K(2') implies that there is an input string u=u u2...ur such that the

stable combinations (X, ul), (sg(X, u U), u2), ..., (sg(X, u1u2.. .uri), ur) are all detectable, and such

that the state xr := sg(X, u) belongs to Aj. Define the intermediate states

X, := Sg(X, Ui), x2 := Sg(X1, U2), ..., Xr = Sg(Xr-1, Ur). (4-1)









As the combinations (xi, ui), i= 1, ..., r, are all stable and detectable combinations, the

states x1, ..., Xr appear as output values of the observer B immediately after having been

reached by Yg. The situation can be depicted as follows.



w

U1U2 ..Ur
Yg: X e A' xr E Aj

Figure 4-1. Equivalence of two asynchronous machines 2' and Y,



The objective of the control unit F is to generate the string u = u u2...ur and apply it as

input to the real machine Y. This action achieves model matching for the present transition for

the following reason. The string u drives the system Yg to the stable state xr, which then

becomes the next stable state of the closed loop system 2c. Then, since h(xr) = h[Ai] = h'(j), the

next stable state of YC produces the same output value as the model 2' to match the model's

response.

Note that the control unit F must operates in a fundamental mode, so the whole system

must have reached a stable combination before the F generates the next input character for Y.

Then, we construct a recursion function 4 for F to implement the above behavior. Keeping in

mind the requirement of fundamental mode operation of the machines, we need to make sure that

the control unit F generates the string u one character at a time and at each step that the

composite system has reached a stable combination before generating the next character. As the

string u has r characters, the control unit F needs r states to accomplish this: 1((x, Ci, w), ..,

r(X, w). The resulting set of states









E(X, wi, w) := {t1(x, Ci, w), ..., (, i, w)}

is associated with the state C' of Y', the state X of E, and the external input character w. To

account for all possible such combinations, the control unit F needs the state set

E := 0 U {Uil,.., UX- A i E(X, 1, Co)},
weA

where o is the initial state of F. We shall use the following notation. For a state x of the

machine Y, let

U(x) := {ae A : sg(x, a) = x

be the set of all input characters that form stable combinations with x. Similarly, for a state C of

the machine 2', let

U'() := {ae A: s'(, a) =

be the set of all input characters that form stable combinations with Q.

Recalling that the control unit is an input/state asynchronous machine F = (A x X, A, E,


0 r, ). The recursion function of F is a function 4:ExXxA-> E and the output function of F

is denoted by r : ExXxA- A. Referring to the configuration (2.24), the output o eX of the

observer B is one of the inputs of F, and the other input is the external input veW. Then, 4

and r are defined as follows.

(i) Let the closed loop system Ec be at a stable combination, where g is at the state X,

namely, the real machine E is at one of the underlying states of the generalized state X. The

observer B has he output value o(=X, and control unit F is at a state e E. Select an element

ceU(X), and define

(,(X,b)) := for all b e U'(),









r(,,(X,a)) := c for all a e A.

This guaranties that the closed loop system 2c and the model X' operate in fundamental mode.

(ii) Suppose that the external input switches to a character w satisfying s'(C', w) = QJ.

Then, the control unit F needs to generate the input string u = uu2...Ur, to take Zg through the

chain of states xl, ..., xr to the state xr e Aj. Meanwhile the output of the observer B will track

the state sequence xl, ..., xr. Thus, the recursion function 4 is defined as follows.

(,(x, w)) := 1(X', ci, W),

1(k(X' i, W),(Xk, w)) := k+l i, w), k = 1, 2, ..., r-1,

r(k((X, w, W),(z, b)) := uk, for any (z,b) e XxA, k = 1, 2, ..., r.

(iii) In response to the last input character ur produced by F, the machine Xg reaches the

desired stable state xr, which implies the real machine X reaches one of the underlying states of

the generalized stable state xr. The machine Xg needs to remain at the state xr until the external

input switches from w to another character. Then, choose an element v e U(Xr) and assign

(('r(X, C, w),(Xr, w)) := r(x, C, W),

r((~r(, C, w),(z, b)) := v, for all (z,b) e XxA.

This completes the construction of the control unit F. Note that whenever the machine Xg

is at a generalized state x, the real machine X is at an corresponding underlying state x' of this

generalized state x and hg(x) = h(x'). This construction achieves model matching of the

generalized machine Xg to the model X' with fundamental model operation as well. This

concludes the proof. +









The proof of Theorem 4-10 includes an algorithm for the construction of a controller C

solving the model matching problem. Then, we use the Algorithm 4.14 in Geng and Hammer

(2005) to build a list A that satisfies condition (ii) of this theorem whenever such a list exists.

This algorithm and Theorem 4-10 give a comprehensive and constructive solution of the model

matching problem. A recursive process is used in the algorithm to build a decreasing chain of

subordinate lists. The last list in this chain, if not deficient, satisfies condition (ii) of Theorem 4-

10; if the last list of the chain is deficient, then there is no controller that solves the requisite

model matching problem.

Let Xg=(A, Y, X, x0, sg, hg) and X'=(A, Y, X', s', h') be the machines of Theorem 4.8, let

E(g, X') = {E1, ..., E'} be their output equivalence list, and let K(X') be the skeleton matrix of

X'. The following steps yield a decreasing chain A(0) >- A(1) >- ... -A(r) of subordinate lists of

E(Xg, X'). The members of the list A(i) are denoted by Al(i), ..., Aq(i); they are subsets of the


state set X of Xg.


Algorithm 4-11 Let -g=(A, Y, X, x0, sg hg) and X'=(A, Y, X', s', h') be the machines of

Theorem 4-10, let E(g X') = {E1, ..., E'} be their output equivalence list, and let K(X') be the

skeleton matrix of X'. The following steps yield a decreasing chain A(0) >- A(1) >- ... >-A(r) of

subordinate lists of E(Xg, X'). The members of the list A(i) are denoted by Al(i), ..., Aq(i); they


are subsets of the state set X of Eg.

Start Step: Set A(0) := E(Xg, X').

Recursion Step: Assume that a subordinate list A(k)= {Al(k), ..., Aq(k)} of E(Xg, X') has

been constructed for some integer k > 0. For each pair of integers ij e {1, ..., q}, let Si(k) be









the set of all states x e A'(k) for which the (ij) element of A(g, AN(k)) is 0; i.e., Sij(k) consists

of all states x E AI(k) for which there is no chain of stable and detectable transitions to a state of

AN(k). Note that Si(k) may be empty. Then set

T ._jSij(k) if Kij(X')=1;
Ti(k): 4 ifKij(X')=0.

Now, using \ to denote set difference, define the subsets

V'(k) := ...,q (k), i = 1, ..., q

A1(k+l) := A(k)\V1(k), i = 1, ..., q

Then, the next subordinate list in our decreasing chain is given by

A(k+l) := { Al(k+l), ..., Aq(k+l)}.

Test Step: the algorithm terminates if the list A(k+l) is deficient or if A(k+l) = A(k);

otherwise, repeat the Recursion Step, replacing k by k+1. *

4.3 A Comprehensive Example of Controller Design

Consider an asynchronous machine X = (A, Y, X, x0, f, h) with the input alphabet A={a, b,

c}, the output alphabet Y={0, 1, 2}, and the state set X={x1, x2, x3, x4}. There is a critical race

pair (x1, c) in the machine (Tables 2-1, Figure 2-1). Let another machine X' = (A, Y, X', Co, f,

h') be the desired model (Table 4-1, Figure 4-2).

The initial state of X is x = x1 and the initial state of X' is C, = i'. After introducing a

burst state x5 = {x2, X3}, we have the generalized machine Yg of X (Table 4-2).











Transition table
a


31


of the machine X'
b c
(2 3
2
43


C)


Figure 4-2. State flow diagram of the machine X'


Table 4-2. Stable state transition table of the machine Y
a b c Y
x' x1 x4 x5 0
X2 x1 X4 X2
x x x x 1
X3 x X3 1
X4 x' X4 2
X5 x X5 1


The initial state of Yg is x = x1. From Table 2-1 and Table 4-2., the output equivalence


list is E(Xg, X') ={E1, E2, E3}, where El={xl}, E2= x2, x3, x5}, E3={x4}. The preliminary


generalized skeleton matrix of the generalized machine is K,(Xg) and the skeleton matrix of the


model is K(X').


Table 4-1.



(2
3









1 0 0 1 1

1 1 0 1 1

Ki(yg) = 1 0 1 1 1

1 0 0 1 1

1 0 0 1 1 -



K(E') = 1 1 1

0 0 1

The subordinate list A of the output equivalence list E(Eg, E') is

Al(1) = {x}, A2(1) = x2, X5, 3(1) = x4}.

The fused skeleton matrix is A(Xg, A) that satisfies

1 1 1 1 1 1

A(g, A)= 1 1 1 1 = K(E')

1 1- 0 0 1

Thus, there exists a controller C to turn the machine E into a deterministic machine that

matches the model E'.

We have a generalized machine g = (A, Y, X, X0, sg, hg) with the state set X ={x, ..,

x5} and a subordinate list A(1) ={A1(1)), A3(1)} that satisfies A(Eg, A(1)) > K(E') and x1

SAl(1). According to the process of construction of the controller C that is described before,

we derive the control unit F and combine it with the observer B of Equation 2-4. Then, we

have the corrective controller C as shown in Figure 2-3.








Recall that the initial state of Y is x = x1 and the initial state of 2' is C = ~1. Maintain

the external input character as a to keep 2' at the state C and maintain the external input

character as a to keep Yg at the state x1. Now we construct F as follows. Set the initial state

of F to o. Denote the states of the observer B by {x x2, x3, x4, x5} and set the initial state of

B to x1. Thus, we have U(x) = {a} and U'(1) = {a} and

F: W(o, (x a)) := o

r&(o, (x1, a)) := a.

B: o(x1, (a, )) = x1 for all Pe Y*.

Assume then the external input character switches from a to b. For the machine Y', s'(l,

b) = C2 and h'(C1) = 0 and h'(C2) = 1, so this transition is detectable. In order to simulate this

transition, the system Yg has to move to a state in A2= {x2, x3, X5}. Since s (x1, c)= x5. For the

real machine Y, it needs to move to either state x2 or state x3 and it does not matter in which

state the Y really stays. In either case, the control unit F needs to generate the character c to

serve as input for Z so that

F: W(o, (x1, b))= $1(xl, C1, b),

r( (xl, C', b), (x1, b)) = c.

B x1,x5 for = 01;
( xl otherwise.

F: 4(5((x1, C', b), (x5, b)) = 1(x1, C', b),

r(S((x1, C', b), (x5, b)) = c.

B: o(x5, (c, P)) = x5 for all PE Y*.









Now consider the other option: the machine Yg is at a stable combination with the state x1

e A1 and X' is at a stable combination with the state (1, when the external input character

switches from a to c. For the machine y', s'((1, c) = C3 and h'((1) = 0 and h'((3) = 2. Thus

this transition is detectable as well. To simulate this transition, the machine Yg needs to move to

a state in A3={x4}, i.e. to x4 So does the real machine X. Since s(x1, b) = x4, the control unit F

needs to generate the character b and this leads to the following

F: (40, (x1, c)) = ((x% c),

r(( (x1, (, c), (x1, c))= b.

Yg: s(x1, b)= x4,

P(x1, b)= h(x)h(x4) = 02

1 ( ) x4 for = 02;
B: y(1, (b, x otherwise.

Then, assume the machine X stays at a stable combination with the state x2 e A2 and the

model is at a stable combination with the state (2, when the external input character switches to

a. The model's response is s'((2, a) =1 so F needs to generate an input character a to drive X

to a state in A1 ={x1}. So we have

F: 4( (xl, C1, b), (x2 a)) = 1(x2, (2, a),

r(((x2, C a), (2 a)) = a.

Yg: s(x2, a)= x1,

P(x2, a)= h(x2)h(x) = 10

B: x, (a, forp 1=0;
B x2 otherwise.









Another possibility is that Y is in a stable combination with the state x3 e A2 and the

model is at a stable combination with the state C2, when the external input character switches

from b to a. The model's response is s'(C2, a) = C so F needs to generate an input character a

to drive Y to a state in A1 ={x1}. Then

F: (5((xl, C1, b), (x3, a)) = 1(x3, 2, a),

r(l(x3, 2, a), (x3, a)) = a.

g : s(x3, a) = x1,

P(x3, a) h(x3)h(x1)= 10

3 x1 for P = 10;
B: o(x3, (a, )) otherwise.

Assume further that Yg is in a stable combination with the state x5 e A2 and the model is

at a stable combination with the state C2, when the external input character switches from b to

a. This case is a combination of the above two cases. Thus, F will generate an input a to drive X

to a state in A1 ={x1}.

F: (5((xl, C1, b), (x5, a)) = 1(x5, C2, a),

r(1(x5, 2, a), (x5, a)) = a.

g : s(x5, a) = x1,

P(x a) = h(x3)h(x) = 10

Sx1 for = 10;
B: o(x (a, x))5= 5 '
: x(a, 13)) otherwise.

This completes the construction of the corrective controller C for the model matching

problem. The state set of the control unit F is

E = {,0, ,1(xl, C1, b), ,1(xl, C1, c), ,1(x2, C2, a), ,1(x3, C2, a), ,1(x5, C2, a)}.










The above state set can be reduced to three states and F can be depicted by Figure 4-3 (the

notation near the arrows indicates the information of (output of the observer B, input of

X,)/output of F).




(x1, b)/c (x1, a)/a (xl, c)/b




(x2, a)/a

(x3, a)/a

(x5, a)/a

Figure 4-3. State transitions diagram of control unit F


The observer B = (A x Y*, X, Z, z0, o, I) is defined according to Equation 2-4 and


described by Figure 4-4. The controller is the combination of F and B according to Figure 2-3.


(c,1)

(c,1) x 3
(a,120)
(a, 120)
(a,O) \


(cO( (a,20)





(a,120)
(c,1) x


(b,2)


Figure 4-4. State transitions diagram of observer B









CHAPTER 5
SUMMARY AND FUTURE WORK

In the present work, the feedback controllers are introduced to correct the faulty behavior

of asynchronous machines. When critical races afflict the asynchronous machine, the existence

of the controllers offer a solution to eliminate the effects of the critical races while controlling

the machine to match a desirable race-free model. We call the problems as Model-Matching

Problems. This approach discloses an interesting and constructive field in which many related

topics are worth investigating.

The solutions have been obtained to the Model-Matching Problem for asynchronous

input/output machines. The concept of generalized state has been used to describe a persistent

state of the machine X about which only partial information is available. The generalized state

allows us to use the partial information available about the state of X to continue controlling the

machine as best as possible toward the goal of achieving model matching, while taking best

advantage of the available information about Y. The results of the Model-Matching Problem

include necessary and sufficient conditions for the existence of the controller, and algorithms for

its construction whenever a controller exists.

The following list is the possible topics for future research:

(i) The algorithm for transforming the generalized stable reachability matrix into skeleton
matrix has been proposed in chapter 3. However, before that we need to raise the power of the
generalized one-step reachability matrix, which requires a large amount of computation. When
the state set of the machine is large, this issue is more significant. If we can obtain a likely one-
step skeleton matrix from the one-step reachability matrix and raise the power of this numerical
skeleton matrix instead, then the calculation is much simpler. But we need spend time to keep all
the information that we need in the transforming and computation.

(ii) The introduction of generalized state transforms an asynchronous machine with critical
races into a deterministic machine. However, the state space is enlarged depending on the
number of critical races. If we can minimize the state space, then it will increase the speed of
computation significantly too.









(iii) Although we can construct an output feedback controller for the closed loop system to
eliminate the effects of critical race whenever a controller exists, this controller may not be
minimal. We can also work on this issue to find out a good strategy to minimize the controller.

(iv) The present discussion excludes the existence of infinite cycles in the existing
machine. We shall deal with the situation when both critical races and infinite cycles occur in the
defective machine.

The controller constructed in the present work ensures that the closed-loop system in

Figure 1-1 and Figure 2-3 operates in fundamental mode. The input changes were only allowed

during stable combinations. This requires the restriction of the controlled machine to those

without any unstable cycles; otherwise the controller can't do anything to correct the machine

once the machine enters a cycle.









LIST OF REFERENCES


Alpan, G. and Jafari, M.A., "Synthesis of a closed-loop combined plant and controller model,"
IEEE Trans. on Systems, Man and Cybernetics, vol. 32, no. 2, 2002, pp. 163-175.

Barrett, G., and Lafortune, S., "Bisimulation, the supervisory control problem, and strong model
matching for finite state machines," Journal of Discrete Event Dynamic Systems, Volume
8, number 4, 1998, pages 377-429.

Chu, T., "Synthesis of hazard-free control circuits from asynchronous finite state machines
specifications," Journal of VLSI Signal Processing, vol.7, no. 1-2, 1994, p. 61-84.

Datta, P.K., Bandyopadhyay, S.K. and Choudhury, A.K., "A graph theoretic approach for state
assignment of asynchronous sequential machines," International Journal ofElectronics,
vol.65, no.6, 1988, p. 1067-75.

Davis, A., Coates, B. and Stevens, K., "The post office experience: designing a large
asynchronous chip," Proceeding of the Twenty-Sihx i Hawaii International Conference on
System Sciences, vol.1, pp. 409-418.

Dibenedetto, M.D., "Nonlinear strong model matching," IEEE Trans. Automatic Control, vol.
39, 1994, pp. 1351-1355.

Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., "Model matching for finite
state machines, Proceedings of the IEEE Conf. on Decision and Control, vol. 3, 1994, pp.
3117-3124.

Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., "Strong model matching for
finite state machines with nondeterministic reference model," Proceedings of the IEEE
Conf. on Decision and Control, vol. 1, 1995, pp. 422-426.

Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., "Strong model matching for
finite state machines, Proceedings ofEuropean Control Conference, vol. 3, 1995, pp.
2027-2034.

Dibenedetto, M.D., Sangiovanni-Vincentelli, A., and Villa, T., "Model matching for finite-state
machines," IEEE Transactions on Automatic Control, vol. 46, no. 11, 2001, pp. 1726-
1743.

Eilenberg, S., "Automata, Languages, and Machines," Academic Press, NY, 1994.

Furber, S.B., "Breaking step the return of asynchronous logic," IEE Review, 1993, pp. 159-162.

Fisher, P.D., and Wu S. F., "Race-free state assignments for synthesizing large-scale
asynchronous sequential logic circuits," IEEE Transactions on Computers, vol. 42, no. 9,
1993, pp. 1025-1034.









Geng, X., "Model matching for asynchronous sequential machines," Ph.D. Dissertation,
Department of Electrical and Computer Engineering, University of Florida, Gainesville,
FL 32611, USA, 2003

Geng, X., Hammer, J., "Input/output control of asynchronous sequential machines, IEEE
Trans. on Automatic Control, Vol. 50, No. 12, pp 1956-1970.

Hammer, J., "On some control problems in molecular biology, Proceedings of the IEEE
Conference on Decision and Control, Vol. 4, 1994, pp. 4098-4103.

Hammer, J., "On the modeling and control of biological signaling chains, Proceedings of the
IEEE Conference on Decision and Control, Vol. 4, 1995, pp. 3747-3752.

Hammer, J., "On corrective control of sequential machines, International Journal of Control,
Vol. 65, No. 65, 1996, pp. 249-276.

Hammer, J., "On the control of incompletely described sequential machines, International
Journal of Control Vol. 63, No. 6, 1996, pp. 1005-1028.

Hauck, S., "Asynchronous design methodologies: an overview," Proceedings of the IEEE, vol.
83, no. 1, 1995, pp. 69-93.

Higham, L. and Schenk, E., "The parallel asynchronous recursion model," Proceedings of the
IEEE Symposium on Parallel and Distributed Processing, 1992, 310-3 16.

Holcombe, W.M.L, "Algebraic Automata Theory, Cambridge University Press, New York,
1982.

Hubbard, P. and Caines, P.E., "Dynamical consistency in hierarchical supervisory control," IEEE
Trans. on Automatic Control, vol. 47, no. 1, 2002, pp. 37-52.

Huffman, D.A., [1954a] "The synthesis of sequential switching circuits, J. Franklin Inst., vol.
257, pp. 161-190.

Huffman, D.A., "The synthesis of sequential switching circuits, J. Franklin Inst., vol. 257,
1954, pp. 275-303.

Huffman, D.A., "The design and use of hazard-free switching networks, Journal of the
Association of Computing Machinery, vol. 4, no. 1, 1957, pp. 47-62.

Isidori, A., "The matching of a prescribed linear input-output behavior in a nonlinear system,"
IEEE Trans. Automatic Control, vol. AC-30, 1985, pp. 258-265.

Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., "Synthesis ofFSMs :
Functional Optimization, Boston, MA: Kluwer, 1997.









Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., "Implicit computation of
compatible sets for state minimization of ISFSMs," IEEE Transactions on Computer Aided
Design, vol. 16, 1997, pp. 657-676.

Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., "Theory and algorithms for
state minimization of nondeterministic FSMs," IEEE Transactions on Computer Aided
Design, vol. 16, 1997, pp. 1311-1322.

Kohavi, Z., "Switching and Finite Automata Theory, McGraw-Hill Book Company, New York,
1970.

Koutsoukos, X.D., Antsaklis, P.J., Stiver, J.A. and Lemmon, M.D., "Supervisory control of
hybrid systems," Proceedings of the IEEE, vol. 88, no. 7, 2000, pp. 1026-1049.

Lavagno, L., Keutzer, K., and Sangicivanni-Vincentelli, A., "Algorithms for synthesis of hazard-
free asynchronous circuits," Proceedings of the 28th ACM/IEEE Conference on Design
Automation, 1991, pp. 302-308.

Lavagno, L., Moon, C. W., and Sangiovanni-Vincentelli, A., "Efficient heuristic procedure for
solving the state assignment problem for event-based specifications, IEEE Transactions
on Computer-Aided Design ofIntegrated Circuits and Systems, Vol. 14, 1994, pp 45-60.

Lin, B. and Devadas, S., "Synthesis of hazard-free multilevel logic under multiple-input changes
from binary decision diagrams," IEEE Trans. on Computer-Aided Design ofIntegrated
Circuits and Systems, vol. 14, no. 8, 1995, pp. 974-985.

Lin, F., "Robust and adaptive supervisory control of discrete event systems, IEEE Transactions
on Automatic Control, vol. 38, nO. 12, 1993, pp. 1848-1852.

Maki, G., and Tracey, J., "A state assignment procedure for asynchronous sequential circuits,"
IEEE Transactions on Computers, vol. 20, 1971, pp. 666-668.

Marshall, A., Coates, B. and Siegel, F., "Designing an asynchronous communications chip,"
IEEE Design & Test of Computers, Vol. 11, no. 2, 1994, pp. 8-21.

Mealy, G.H., "A method for synthesizing sequential circuits," Bell System Tech. J., vol. 34,
1955, pp. 1045-1079.

Moon, C.W., Stephan, P.R., and Brayton, R.K., Synthesis of hazard-free asynchronous circuits
from graphical specifications," IEEE International Conference on Computer-Aided
Design, 1991, pp. 322 -325.

Moore, B. and Silverman, L., "Model matching by state feedback and dynamic compensation,"
IEEE Trans. Automatic Control, vol. 17, 1972, pp. 491-497.

Moore, E.F., "Gedanken-experiments on sequential machines," Automata Studies, Annals of
Mathematical Studies, no. 34, Princeton University Press, N.J., 1956.









Moore, S.W., Taylor, G.S., Cunningham, P.A., Mullins, R.D. and Robinson, P., "Self-calibrating
clocks for globally asynchronous locally synchronous systems," Proceedings ofInter.
Confer. on Computer Design, 2000, pp. 73-78.

Morse, A.S., "Structure and design of linear model following systems", IEEE Trans. Automatic
Control, vol. 18, 1973, pp. 346-354.

Murphy T.E., Geng X., and Hammer J., "Controlling races in asynchronous sequential
machines," Proceeding of the IFAC World Congress, Barcelona, July 2002..

Murphy T.E., Geng X., and Hammer J., "On the control of asynchronous machines with races,"
IEEE Transactions on Automatic Control, vol. 48, no. 6, pp. 1073-1081.

Murphy, T.E., "On the control of asynchronous sequential machines with races, Ph.D.
Dissertation, Department of Electrical and Computer Engineering, University of Florida,
Gainesville, FL 32611, USA, 1996.

Nishimura, N., "Efficient asynchronous simulation of a class of synchronous parallel
algorithms," Journal of Computer and System Sciences, vol. 50, no. 1, 1995, pp. 98-113.

Nowick, S.M., "Automatic synthesis of burst-mode asynchronous controllers," Ph.D.
Dissertation, Stanford University, 1993.

Nowick, S.M., and Coates, B., "UCLOCK: automated design of high-performance unclocked
state machines," ICCD '94. Proceedings., IEEE International Conference on Computer
Design, 1994, pp. 434-441.

Nowick, S.M., Dean, M.E., Dill, D.L. and Horowitz, M., "The design of a high-performance
cache controller: a case study in asynchronous synthesis," Proceedings of the 26th Hawaii
International Conference on System Sciences, 1993, pp. 419-427.

Nowick, S.M. and Dill, D.L., "Synthesis of asynchronous state machines using a local clock,"
IEEE International Conference on Computer Design, 1991, pp. 192-197.

Oliveira, D.L., Strum, M., Wang, J.C. and Cunha, W.C., "Synthesis of high performance
extended burst mode asynchronous state machines," Proceedings. 13th Symposium on
Integrated Circuits and Systems Design, 2000, pp. 41-46

Ozveren, C.M., Willsky, A.S., and Antsaklis, P.J., "Stability and stabilizability of discrete event
systems, J ACM, Vol. 38, 1991, pp. 730-752.

Park, S.-J. and Lim, J.-T., "Robust and nonblocking supervisory control of nondeterministic
discrete event systems using trajectory models, IEEE Trans. on Automatic Control, vol.
47, no. 4, 2002, pp. 655-658.

Peterson, J.L., "Petri Net Theory and The Modeling of Systems, Prentice-Hall, NJ, 1981.









Ramadge, P.J.G., and Wonham, W.M., "supervisory control of a class of discrete event
processes," SIAM Journal of Control and Optimization, vol. 25, no. 1, 1987, pp. 206-230.

Ramadge, P.J.G., and Wonham, W.M., "The control of discrete event systems," Proceedings of
IEEE, vol. 77, no. 1, 1989, pp. 81-98.

Cole, R. and Zajicek, O., "The expected advantage of asynchrony," Journal of Computer and
System Science, vol. 51, no. 2, pp. 286-300.

Shields, M.W., "An Introduction to Automata Theory, Blackwell Scientific Publications,
Boston, 1987.

Thistle, J. G. and Wonham, W.M., "Control of infinite behavior of finite automata, SIAM
Journal on Control and Optimization, vol. 32, no. 4, 1994, pp 1075-1097.

Unger, S. H., "Asynchronous Sequential Switching Circuits, Wiley-Interscience, New York,
NY, 1969.

Unger, S. H., "Self-synchronizing circuits and non-fundamental mode operation," IEEE Trans.
Computers, vol. 26, no. 3, 1977, pp. 278-281.

Unger, S. H., "Hazards, critical races, and metastability, IEEE Trans. on Computers, vol. 44,
no. 6, 1995, pp 754-768.

Venkatraman, N., "On the control of asynchronous sequential machines with infinite cycles, "
Ph.D. Dissertation, Department of Electrical and Computer Engineering, University of
Florida, Gainesville, FL 32611, USA, 2004

Venkatraman, N., "On the control of asynchronous sequential machines with infinite cycles, "
International Journal of Control, Vol. 79, No. 07, 2006, pp. 764-785.

Yu, M.L. and Subrahmanyam, P.A., "A path-oriented approach for reducing hazards in
asynchronous designs," Proceedings of Design Automation Conference 29th ACM/IEEE,
1992, pp. 239 -244.









BIOGRAPHICAL SKETCH

Jun Peng was born in Wuhan, Hubei Province, China. She received her bachelor's degree

in automatic control and master's degree in control theory and control engineering from

Shanghai Jiao Tong University, Shanghai, China, in July 2000 and in April 2003, respectively.

She began her Ph.D. program in the Department of Electrical and Computer Engineering at

University of Florida, Gainesville, FL in August 2003. Her research interests include

asynchronous sequential circuits, application of asynchronous sequential systems in computer

architecture, artificial intelligence and biological systems, control theory, control systems, and

applications of control theory in computer communication networks. She received her Ph.D. in

August 2007.





PAGE 1

1 INPUT/OUTPUT CONTROL OF ASYNC HRONOUS MACHINES WITH RACES By JUN PENG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

PAGE 2

2 2007 Jun Peng

PAGE 3

3 To Mom and Dad.

PAGE 4

4 ACKNOWLEDGMENTS I thank my advisor, Dr. Jacob Hammer, fo r his excellent, professional guidance and support during my four years at th e University of Florida. I th ank Dr. John Schueller, Dr. Haniph Latchman, and Dr. Janise McNair, for serving on my Ph.D. supervisory committee and for their continuous help and advice. I thank my officemates, Debraj Chakraborty and Niranjan Venkatr aman, for interesting and joyful discussions. I thank all my friends for their care and friendship. Last but not least, I thank my brother and his wife for their understandin g, support and love throughout my school years. I thank my pa rents, Mingqing Peng and Mingxia Wu, whose unceasing love and whole-hearted support made finishing this work possible. I thank my husband, Zhipeng Liu, for his love and faith in me.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........6 LIST OF FIGURES................................................................................................................ .........7 ABSTRACT....................................................................................................................... ..............8 1 INTRODUCTION................................................................................................................... .9 2 TERMINOLOGY AND BACKGROUND............................................................................14 2.1 Asynchronous Sequential Machines...............................................................................14 2.2 Generalized Machines, States and Functions.................................................................21 2.3 Observer................................................................................................................. .........27 3 REACHABILITY OF A GENERALIZED MACHINE........................................................31 3.1 Generalized Reachability Matrix....................................................................................31 3.2 Common-output Generalized States...............................................................................41 3.3 Output Feedback Trajectory...........................................................................................44 3.4 Preliminary Generalized Skeleton Matrix......................................................................51 4 MODEL MATCHING FOR INPUT/OUT PUT ASYNCHRONOUS MACHINES WITH RACES..................................................................................................................... ...56 4.1 Model Matching Problem...............................................................................................56 4.2 Existence of Controllers.................................................................................................60 4.3 A Comprehensive Exampl e of Controller Design..........................................................67 5 SUMMARY AND FUTURE WORK....................................................................................74 LIST OF REFERENCES............................................................................................................. ..76 BIOGRAPHICAL SKETCH.........................................................................................................81

PAGE 6

6 LIST OF TABLES Table page 2-1 Transition table of the machine .....................................................................................17 2-2 Transition table of the machine s.....................................................................................17 3-1 Stable state transition table of the machine g..................................................................33 3-2 Transition table of the machine .....................................................................................43 3-3 Transition table of the machine s....................................................................................43 4-1 Transition table of the machine ....................................................................................68 4-2 Stable state transition table of the machine g..................................................................68

PAGE 7

7 LIST OF FIGURES Figure page 1-1 Control configuration for the asynchronous machine ...................................................10 2-1 State flow diagram of the machine ................................................................................17 2-2 State flow diagram of the machine s...............................................................................18 2-3 Control configuration for the closed-loop system c.........................................................28 3-1 State flow diagram of the machine g...............................................................................33 4-1 Equivalence of two asynchronous machines and g..................................................63 4-2 State flow diagram of the machine ...............................................................................68 4-3 State transitions diag ram of control unit F........................................................................73 4-4 State transitions di agram of observer B............................................................................73

PAGE 8

8 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INPUT/OUTPUT CONTROL OF ASYNC HRONOUS MACHINES WITH RACES By Jun Peng August 2007 Chair: Jacob Hammer Major: Electrical and Computer Engineering The occurrence of races causes unpredictable and undesirable behavior in asynchronous sequential machines. In the pres ent work, traditional feedback control techniques are used to control a race-afflicted machine, so as to turn it into a deterministic machine that matches a desired model. Instead of replaci ng or redesigning the whole machine, I add an output feedback controller to the original defective machine, and the controller eliminates the negative effects of the critical races. The present work focuses on asynchronous sequential machines in which the state of the machine is not provided as an output. The results include the necessary and sufficient conditions for the existence of controllers that eliminate the effect s of a critical race, as well as al gorithms for their design. The necessary and sufficient conditions for the existence of contro llers are presented in terms of certain matrix inequalities.

PAGE 9

9 CHAPTER 1 INTRODUCTION Asynchronous sequential machines are digital logic circuits without synchronizing clock, so they are also called clockless logic circui ts. The lack of a synchronizing clock allows asynchronous machines to operate much faster. In addition, there are practical applications, such as parallel computation, where the underlying system is inherently asynchronous. Asynchronous design techniques can also be used to achieve ma ximum efficiency in parallel computation (Cole and Zajicek (1990), Higham a nd Schenk (1992), Nishimura (1995 )). The design of asynchronous machines has been an active area of research since mid 1950s (Huffman (1954, 1955)). Potential difficulties, such as critical races and infinite cycles that may arise in the design of asynchronous machines are discussed in the litera ture (Kohavi (1970), Unger (1959)). Both critical races and infinite cycles are flaws in the operation of an asynchronous sequential machine. In this dissertation, I w ill focus on asynchronous machines with critical races. A critical race drives the machine to ex hibit nonpredictable behavior, and it may be caused by malfunctions, by design flaws, or by implementa tion flaws. Common practice is to rebuild a machine that is afflicted by a critical race, and replace it with a race-free machine. In the present work, I use tradit ional feedback control techniqu es to control a race-afflicted machine, so as to turn it into a deterministic machine that matches a desired model. Instead of replacing or redesigning a defective machine, I add an output feedback controller, and the controller eliminates the effects of the critical races. The feedb ack controller turns the closed loop system into a deterministic machine, and th e closed loop system imitates the desired model (Figure 1-1).

PAGE 10

10 Figure 1-1. Control configuration for the asynchronous machine Here, is the machine being controlled and C is another asynchronous machine that serves as an output feedback controller. We denote by c the machine described by the closed loop. The objective is to fi nd a controller C for which the closed loop machine c exhibits desirable behavior. I represent the behavior desired for the clos ed loop system by an asynchronous machine called a model. In thes e terms, the objective is to design a controller C for which the closed loop machine c simulates the model Of course, the machine that represents the desired behavior is not afflicted by any critical races. Thus, by si mulating the behavior of the closed loop system eliminates the ill effects of the critical ra ces present in The problem of designing such a controller C is often referred to as th e model-matching problem. The objective, then, is to find necessary and sufficient conditions for the existence of the controller C that solves the model matching problem. When such a controller exists, I also provide an algorithm for its construction. The literatures regarding the model-matchi ng problem of asynchronous machines with races has focused so far on asynchronous sequential machines in which the state is provided as the output of the machines (input/state machines). In Murphy et al. (2002, 2003), the control of asynchronous machines was discussed and state feedb ack controllers that e liminate the effects of critical races in asynchronous machines were developed. In Venkatraman and Hammer (2004), C u y v

PAGE 11

11 state feedback controllers were used to eliminate the effects of infinite cycles on asynchronous machines; and in Geng and Hammer (2005), th e problem of model matching with output feedback controllers was considered for asynchr onous machines with no cr itical races. In the present work, I concentrate on the problem of desi gning output feedback controllers that eliminate the effects of critical races in asynchronous machines. The problem of eliminating the effects of critical races with output feedback requires the development of additional notions as well as the development of new design algorithms for controllers, and these are the subjects of our present discussion. To introduc e these notions, I start with a brief review of some of the underpinnings of the theory of asynchronous machines. Unlike a synchronous machine, which is driven by clock pulses, an asynchronous machine is driven by changes of its input variables. A stable state is a st ate at which the machine lingers until a change occurs in one of its input variables. In general, the change of an input variable causes an asynchronous machine to go through a succession of state transitions. If this succession of transitions ends, then the final stat e reached by the machine is a stable state; the states through which the machine passes during the succession are unstable states. Ideally, an asynchronous machine passes through an unstable stat e in zero time. Thus, unstable states are not noticeable to the user. If an asynchronous machine has a succession of st ate transitions that does not terminate, then the machine has an infinite cycle. Infinite cycles form another class of potential defects of an asynchronous machine. The elimination of the effects of infinite cycles by the use of state feedback was discussed in Venkatraman a nd Hammer (2004). Asynchronous machines with infinite cycles are not discu ssed in this dissertation; I a ssume that all machines under consideration have no infinite cycles.

PAGE 12

12 To guaranty the proper behavior of an as ynchronous machine, some care has to be exercised during its operation. In pa rticular, one has to avoid cha nging values of input variables while the machine undergoes a succession of state transitions. If an input change occurs while the machine is not in a stable state, then, due to asynchrony, it is not possi ble to predict the state of the machine at the instant in which the input change occurs. As the response of the machine depends on its state, this may result in an unpredic table response of the machine. In other words, the response may vary depending on the specific state of the machine at the instant of the input change. To avoid this situation, asynchronous m achines are normally operated so as to guaranty that input changes occur only while the machine is in a stable st ate. When this precaution is taken, we say that the machin e operates in fundame ntal mode. In this dissertation, all asynchronous machines are operated in fundamental mode. The development of necessary and sufficien t conditions for the existence of a model matching controller C and the algorithm for its construction depend on a certain generalized concept of state, introduced in ch apter 2 below. A generalized state describes a persistent state of the machine about which only partial information is available. More specifically, as is an input/output machine, it is not always possible to determine its current state from available input/output data. A generalized state indicate s a situation in which it is known that is in a stable state, but the exact state of is not known; the machine can be in any one of a predetermined set of stable states. The generalized state allows us to use the partial information available about the state of to continue controlling the mach ine as best as possible toward the goal of achieving model matching, while taking be st advantage of the available information about The generalized state allows us to formaliz e in a concise and functional way the future implications of uncertainties in the present state of the machine

PAGE 13

13 The notion of a generalized state was also used in Venkatraman and Hammer (2004) to represent phenomena related to the presence of infi nite cycles. In the present paper, I show that a generalized notion of state can also be used to represent uncertainty in asynchronous machines with critical races, in situa tions where the exact state of the machine is not known. The mathematical background of our discussi on is based on Eilenberg (1974). Studies dealing with other aspects of the control of sequential machin es can be found in Ramadge and Wonham (1987) and in Thistle a nd Wonham (1994), where the theory of discrete event systems is investigated; in Ozveren, Willsky, and Antskl is (1991), where stability issues of sequential machines are analyzed; and in Hammer ( 1994, 1995, 1996a and b, 1997), Dibenedetto, Saldanha, and Sangiovanni-Vincentelli (1994), Barrett an d Lafortune (1998), where issues related to control and model matching for sequential machin es are considered. Th ese discussions do not take into consideration specialized issues related to the f unction of asynchronous machines, like the issues of stable states, unstable states, and fundament mode operation. As a result, these works refer mostly to the c ontrol of synchronous machines.

PAGE 14

14 CHAPTER 2 TERMINOLOGY AND BACKGROUND 2.1 Asynchronous Sequential Machines Definition 2-1 An asynchronous sequential machine is defined by a sextuple (A, Y, X, x0, f, h), where A, Y and X are nonempty finite sets: A is the input set, Y is the output set, and X is the state set. x0 X is the initial state of the m achine. The partial function f : A X X is the state transition function (or recurs ion function) and the partial function h : A X Y is the output function. When the output function h does not depend on th e input character (i.e., when h : X Y), the machine is called a Moore machine in Moore (1956). Note that every asynchronous machine can be represented as a Moore machine. The machine operates according to a recursion form xk+1 = f(xk,uk) yk = h(xk) k = 0 1 2 ... (2-1) Where, k counts the steps of the machine The sequences xk, uk, and yk are the state sequence, the input sequence and the output sequence, respectively. The machine is an input/state machine if Y = X, or yk = xk for each step k 0. When the output is not the state, then the machine is an input/output machine. The present paper focuses on input/output machines. Definition 2-2 Let be an asynchronous machine repr esented by sextuple (A, Y, X, x0, f, h). A pair (x, u) X A is called a valid pair if the recursi on function f is defined at it. If x = f(x, u), then the combination ( x, u) is a stable combination. Definition 2-3 Let (x, u) be a vali d pair of the machine = (A, Y, X, x0, f, h). If (x, u) is not a stable combination, then the machin e generates a chain of transitions x1 = f(x, u), x2 = f(x1,

PAGE 15

15 u), If the chain does not te rminate, then the machine contains an infinite cycle. If the succession of transitions ends at a stable combination (xi, u), then xi is the next stable state of x with the input character u. In the present work, I assume that none of our asynchronous machines possess infinite cycles. Definition 2-4 Let Y be an alphabet and let y1, yq Y be a list of characters such that yi+1 yi for all i = 1, q 1. Then, the burst of a string y = y1y1 y1y2y2 y2 yqyqyq is (y) : y1 y2 yq-1 yq and -1(y) y1y2 yq-1, for q > 1; -1 for q = 1. Let x1x2x3 xm be the string of states generated by the machine from valid pair (x, u) and xm is the next stable state. Then, the burst of the valid pair ( x, u) is defined as (xm, x, u) : (h(x)h(x1)h(x2)h(xm-1)h(xm)). Definition 2-5 Let p1 and p2 be two strings of the alphabet A. As usual, p2 is a prefix of p1 if there is a string p3 such that p1 = p2p3. We say that p2 is a strict prefix of p1 if p3 the empty string. For example, given three strings p1, p2, and p3: p1 = y1y1y2y3; p2 = y1y1y2y3; and p3 = y1y1y2y3y2y3. The string p1 and p2 are each others prefix string. Both p1 and p2 are strict prefix strings of the string p3. Definition 2-6 A state-input pair (r, v) for which the next stable state of the machine is unpredictable is called a cr itical race pair, or, brie fly, a critical race. There may be more than one possible next stable states of the critical race pair (r, v). Let us suppose there are m possible next stable stat es of (r, v). The set of all these states {r1, r2,

PAGE 16

16 rm} is called the outcomes of the race. Correspondingl y, there are m bursts for the critical race pair (r, v), one for each possible outcome of the race. Let i be the burst generated by the machine when the outcome of the race is ri. Then, we refer to the set (r, v) : { (r1, r, v), (r2, r, v) (rm, r, v)} as the burst set of the critical race (r, v). More details about races of asynchronous machines can be found in Kohavi (1970). Definition 2-7 For a deterministic asynchronous machine = (A, Y, X, x0, f, h), let x be the next stable state of a valid pair (x, u). The stab le recursion function s : X A X of is given by s(x, u) : x for all valid pairs (x, u) X A. The stable state machine induced by is represented by the sextuple |s = (A, Y, X, x0, s, h). For an asynchronous machine with a critical race pair (r, v), the stable recursion function s has multiple values at the pair (r, v), say, s(r, v) : {r1, r2, rm}. Here, r1, r2, rm are the outcomes of the critical race (r, v). Definition 2-8 Let Xr := {xi(1), ..., xi(m)} be a set of states of the machine and assume they have a non-empty set U of comm on input characters. For an element u U, let s[Xr, u] be the set of all possible next stable states, wh ere s is the stable transition function of the machine Let B(Xr,u) be the set of all bursts from Xr to s[Xr, u]. For each burst B(Xr, u), let X( ) s[Xr,u] be the set of all states x s[Xr, u], i.e., the set of all states that can be reached from Xr via the burst We refer to X( ) as a burst equivalent set for the burst with respect to input u. Note that the burst equi valent sets in s[Xr, u] may not be disjoint. The following is an example to show how to obtain the stable state machine and transition equivalent set for a given subset of its state set. Consider a machine with the input alphabet

PAGE 17

17 A={a, b, c}, the output alphabet Y={0, 1, 2}, and the state set X={x1, x2, x3, x4 }. There is a critical race pair (x1, c) in the machine. This machine is depicted by a chart (Table 2-1) or a figure (Figure 2-1). So is the stable state machine s of (Table 2-2, Figure 2-2). Table 2-1. Transition table of the machine a b c Y x1 x1 x4 {x2, x3} 0 x2 x4 x1 x2 1 x3 x4 x3 1 x4 x1 x4 2 x3 c a x1 x2 x4 c b b a b a c a c Figure 2-1. State flow diagram of the machine Table 2-2. Transition table of the machine s a b c Y x1 x1 x4 {x2, x3} 0 x2 x1 x4 x2 1 x3 x1 x3 1 x4 x1 x4 2

PAGE 18

18 x3 c a x1 x2 x4 c a b b b a c a c Figure 2-2. State flow diagram of the machine s Let Xr = {x1, x2, x3}, then U = {a, c}. Given u = a, the next stable states set is s[Xr, a] = {x1}. The burst set is (Xr, u) = { 2} = {0, 020} and the two burst equivalent subsets of s[Xr, a] are: S1 = {x1} and S2 = {x1}. The state in S1 can be reached via burst 1=0 and the state in S2 can be reached via burst 2=020, so S1 and S2 are two burst equivalent sets. From the above example, we notice that thos e states in a burst equivalent set of an asynchronous machine cannot be distinguished from each other by an external observer. So, we need a new method to deal with this kind of situation. Definition 2-9 Let Xr := {xi(1), ..., xi(m)} be a set of states of the machine assume they have a non-empty set U of comm on input characters, and let u U be a character. The asynchronous machine = (A, Y, X, x0, f, h) is detectable at the set pair (Xr, u) if it is possible to determine from input/output data whether all outcomes s[Xr, u] have reached their next stable state; if so, the set of transitions from (Xr, u) to s[Xr, u] is called a stable and detectable transition set.

PAGE 19

19 It was shown in Geng and Hammer (2005) that a stable transition is detectable if and only if its burst switches output characters in its last step. When dealing with de tectability of sets of states, the situation is somewhat more complicat ed. The outcomes of a st ate-set-input pair (Xr, u) form a set of states s[Xr, u] and all bursts from the set Xr to s[Xr,u] form a set of bursts B(Xr, u). Even if every burst in B(Xr, u) is detectable individually, it is still possible that one cannot determine whether the machine has reached its next stable state. For instance, consider two bursts in (Xr, u): (x1, Xr, u) and (x2, Xr, u). The burst (x1, Xr, u) is a strict prefix of the burst (x2, Xr, u), say (x1, Xr, u) = y1y2y3 and (x2, Xr, u) = y1y2y3y4y5y3. Clearly, then, at the point where the burst (x1, Xr, u) ends, it is not possible to te ll whether the machine has reached its next stable combination, as the machine might actually be on its way to the state x2. This discussion leads us to the following statement. Proposition 2-10 Let Xr := {xi(1), ..., xi(m)} be a set of states of the machine and assume they have a non-empty set U of co mmon input characters. For an element u U, let s[Xr, u] be the set of all possible next stable states, where s is the stable transition function of the machine Let B(Xr, u) be the set of all possible bursts generated by the pair (Xr, u). The asynchronous machine is detectable at the set pair (Xr, u) if and only if the following conditions are satisfied: (a) -1 for all B(Xr, u); (b) is not a strict prefix of for any (Xr, u). Proof The first condition has been proved in Geng and Hammer (2005). Let us examine the second condition. The first condition guarantees the detectability of the end of each burst in B(Xr, u), so the only confusion is fr om other bursts in the set B(Xr, u). Consider two bursts i, j

PAGE 20

20 B(Xr, u), where i leads to the state xi, while j leads to the state xj. Assume first that is detectable at the pair (Xr, u). By contradiction, assume that i is a strict prefix of j, i.e., i = y1y2yk-1yk and j = y1y2yk-1ykyk+1y where, yk yk-1 and y y-1. Once the change from yk-1 to yk is observed, it is not possible to determine whether the machine has reached the next stable state xi or whether it is still in the transition to next stable state xj. Thus the machine cannot be detectable at (Xr, u), a contradiction. This shows that condition (b) must be valid whenever is detectible at (Xr, u). Conversely, assume that conditions (a ) and (b) are both valid, and let (Xr, u) be a burst, and let X s[Xr, u] be the set of all st ates to which the burst leads. Now, since is not a strict prefix of any other burst in B(Xr, u), it follows that, at the end of the burst the machine must be at one of the states of the set X say the state x (i.e., cannot be on its way to other states). Furthermore, since the end of the burst can be determined, and whence it can be determined that has reached a stable comb ination with a state of X (note that it cannot be determined from the burst which state of X has been reached). This completes our proof. For example, consider the machine with transition tabl e of Table 2-1, let Xr = {x1, x2}, then U = {a, b, c}. Let us check the detectability of the combination (Xr, a). The next stable states set s[Xr, a] = {x4}. The burst from the state x1 to the state x4 is 1 = 02 and the burst from the state x2 to the state x4 is 2 = 102. The burst set x4, Xr, a) = { 1, 2}. Since the burst 1 is not a strict prefix of 2, and vice versa, the transition from (Xr, a) to x4 is detectable.

PAGE 21

21 2.2 Generalized Machines, States and Functions The next notion is central to our discussion. It is sometimes convenient to consider certain sets of states of a machine as one quantity. This is c onvenient, for example, in cases where the available data at a certain point in time does not permit us to distinguis h between these states. This leads us to the following notion of a generalized machine. Definition 2-11 Let be a machine with the state se t X and input set A, let S( ) be a burst equivalent set with respect to u of the machine containing more than one state, and let be a set disjoint from X, and let : P(X) be a function. Associate with S( ) the element xb := (S( )); we call xb a burst state. The set is then called the set of potential burst states and is called the burst state assignment f unction. Let A be the set of all common input characters of the states in S( ). Then, the set of a ll valid pairs of xb is given by {( xb, a) : a A}. The set A is also called the valid input set of the burst state xb. Let Xb be the set of all burst states of the machine The generalized state set X of is the union X Xb. The burst equivalent set S( ) represented by a burst state xb is also recorded as S(xb). Let |s = (A, Y, X, x0, s, h) be the stable state machine of an asynchronous machine Let Xr := {xi(1), ..., xi(m)} X be a set of states of the machine and assume they have a nonempty set U of common input characters. For an element u U, let s[Xr, u] be the set of all possible next stable states. Let (Xr, u) { 1, 2, } be the set of all possible bursts generated by the transition (Xr, u) s[Xr, u], and let S( 1), S( 2), S( ) be the burst equivalent sets in s[Xr, u]. For each i = 1, 2, ..., we distinguish between two cases: 1) The set S( i) contains a single state x X. Then, we identify S( i) with the state x.

PAGE 22

22 2) The set S( i) contains more than one stat e. Then, we associate with S( i) a burst state, which represents the fact that these stat es are indistinguishabl e in this transition. For the second case, let Ai be the set of all common input characters of S( i), i = 1, 2, Note that Ai cannot be empty, since at least Ai contains the element u. Definition 2-12 Let = (A, Y, X, x0, f, h) be an asynchronous machine with a generalized state set X = Xb X, where X is the regular state set of and Xb is the burst state set of the machine We build now a generalized stable transition function sg : X A P( X) as follows: 1) For all states x X and all input characters u A, set sg(x, u) : s(x, u). 2) For a burst state x Xb, let U(x) A be the set of all input characters that form valid pairs with x. Let S(x) be the burst equivalent set represented by the burst state x. For an input character a U(x), let 1, 2, m be the set of all burst equivale nt subsets of s[S(x), a]. If i contains more than one state, then let xi be the burst state associated with the set i; otherwise i is represented by its only state xi, i = 1, ..., m. Then, sg(x, a) : {x1, ..., xm}. Note that the next stable states of a mach ine can be a combination of burst states and regular states. In the next discussion, I will give a more speci fic algorithm to build th e generalized stable transition function. As a practical process, th e algorithm should avoid getting involved into infinite loops. Thus, before the construction we need to make sure about two issues: a) the process of building the generalized stable transiti on function includes finite steps; b) there is no infinite cycles created in the construction. Since ever y burst equivalent set S( ) is a subset of the state set X, for an asynchronous machine with n regular stat es, the maximum number of

PAGE 23

23 subsets in X is 2n. Hence, the number of burst equiva lent sets is equal or less than 2n. Namely, the number of burst stat es generated in the machine is finite. Then the first requirement is guaranteed. In the previous di scussion, we have excluded asynchronous machine with infinite cycles. So, any transition starting from a regular state of a machine in this paper ends at the next stable states. Similarly, unde r the definitions of the generalized state and generalized stable transition function, for each vali d state-input pair there is one or more next stable states. Hence, each transi tion starting from a generalized st ate ends at the next stable states, i.e., no infinite cycles will be created in the process of defining the generalized stable transition function sg. Consider an asynchronous machine (A, Y, X, x0, f, h) with stable state machine |s = (A, Y, X, x0, s, h). Let X = Xb X be the generalized state set and let Xb : { 1, 2, t} be the burst state set of For every burst state c Xb, let Ab : {a1, a2, ag(c)} be the valid input set of c. For every valid pair ( a), Xb and a Ab, let S( ) be the set of regular states represented by and let s[S( a) be the set of all possibl e next stable st ates of [S( ), a]. Assume that |s have critical races (r1, v1), (r2, v2), (r, v), and let T(ri, vi) := {r1 i, r2 i, rm(i) i} X be the set of all outcom es of the critical race (ri, vi), i =1,, We build the generalized stable transition function sg with the following algorithm. Algorithm 2-13 Consider an asynchronous machine (A, Y, X, x0, f, h) with stable state machine |s = (A, Y, X, x0, s, h). Let X = Xb X be the generalized state set, where Xb is the burst state set and X is the regular state set. Assume that |s has critical races (r1, v1), (r2, v2), (r, v), and let T(ri, vi) := {r1 i, r2 i, of the rm(i) i} X be the set of all outcomes

PAGE 24

24 critical race (ri, vi), i =1,, For every state x X and u A, if s(x, u) is a single state, then set sg(x, u) : s(x, u). Set Xb := and let (r1, v1), (r2, v2), (r, v) be the set of all cr itical race pairs of the machine Set i := 1 and run the following steps: Step 1 : a) Consider the i-th element (ri, vi) of the set K. If i then let Xi := {r1 i, r2 i, rm(i) i} be the outcomes of the critical race (ri, vi). b) If i > then the i-th element (ri, vi) of the set K is a burst -state-input pair created in Step 3 of a previous cycle of the algorithm. Let S(ri) be the state set associated with the burst state ri. Let Xi := s[S(ri), vi] be the set of all possible next stab le states of the set of states S(ri) with the input character vi. Step 2 : Set j = 0. Partition the set Xi into its burst eq uivalent subsets T1, T2, Tt with respect to the input character vi, and denote by T := {T1, T2, Tt} the corresponding class of subsets. Let Z be the set cons isting of all subsets Tj that contain a single st ate; if there are no such subsets in T, then set Z := Denote by Si := T \ Z the corresponding difference set. If Si = then set k := 0 and go to Step 4. Otherwise, Let Si 1, Si 2, Si k be the members of Si. Step 3 : Set j:= j+1 and check the set Si j as follows. Let be the set of poten tial burst states and let : P(X) be the burst state assignment function. If (Si j) Xb, then proceed as follows; otherwise, go to b).

PAGE 25

25 Add the burst state xi j := (Si j) to Xb, i.e., set Xb := Xb xi j. Let Ac : {u1, u2, ug(i,j)} be the valid input set of the burst state xi j. Let := #K be the number of elements of the set K. Add to K the elements (r, v) := (xi j, u), = 1, ..., g(i,j). Add the burst state (Si j) to Z. If j < k, then go back to Step 3. Step 4 Set sg(xi, ui) : Z. Step 5. If i < #K, then set i = i + 1 and go back to step 1. Otherwise, terminate the algorithm. The set X : X Xb is the generalized st ate set of the machine The generalized stable transition function is sg : X A P( X). According to definition of the burst of a string the last character of a burst is the output value of the system for the corres ponding state. Consequently, all st ates in a burst equivalent set have the same output value. This implies that the following is true. Lemma 2-14 The output value of a burst state x is the output value of any state in the corresponding burst equivalent set S(x). Definition 2-15 Let = (A, Y, X, x0, f, h) be an asynchronous machine with a generalized state set X = Xb X, where X is the regular state set of and Xb is the burst state set of the machine Let x be a generalized state of the machine The generalized output function hg : X A Y of is defined as follows: 1) For all states x X, set hg(x) h(x);

PAGE 26

26 2) For all burst state x Xb, let S(x) be the burst equivalent set that is associated with the burst state x. Set hg(x) h(x ), x S(x). For example, consider the machine with transition table of Table 2-1, which has one critical race s(x1, c) = {x2, x3}. Using Algorithm 2-13, we can get the burst state set Xb = {x5} and x5 represents the subset {x2, x3}. The generalized stable recursion function sg and the generalized output function hg of the machine can also be defined (Table 2-3). Table 2-3. Stable transition table of the generalized a b c Y x1 x1 x4 x5 0 x2 x1 x4 x2 1 x3 x1 x3 1 x4 x1 x4 2 x5 x1 x5 1 Definition 2-16 Let = (A, Y, X, x0, f, h) be an asynchronou s machine with the stable state machine |s = (A, Y, X, x0, s, h). Then, g = (A, Y, X, x0, sg, hg) is the generalized machine associated with where X is the generalized state set, sg is the generalized stable recursion function, and hg is the generalized output function of the machine When an asynchronous machine is enhanced into a generalized machine, it still keeps some properties. We address two prope rties of the generalized machine in these two statements. Lemma 2-17 Given an asynchronous machine with a state set X, which contains finite number of states. Then the a ssociated generalized machine g also has a generalized state set X with finite number of states.

PAGE 27

27 Proof Let us suppose the machine has a state set X = {x1, x2, xn} and the generalized machine g has a generalized state set X = {x1, x2, xn+t} From the definition of burst equivalent set, a ny burst equivalent set S( ) is a subset of the state set X. Then, the maximum number of burst equivalent sets cannot be larger than the number of subsets of X, i.e., 2n, and hence is finite. Lemma 2-18 If the machine has no infinite cycles, neither does the machine g. Proof. Assume the machine has no infinite cycles but the generalized machine g, which is derived from the machine has one infinite cycle of length i, where i > 1. Suppose that i generalized states x1, x2, ..., xi are involved in this infinite cycle The states x1, x2, ..., xi may be regular states or burst states. Let us consider the followi ng two cases: i) If all these i states are regular states. Then it implies that the machine has at least one infinite cycle And the infinite cycle invol ves the i states x1, x2, ..., xi of the machine t conflicts with the assumption that the machine has no infinite cycles. ii) If in the i states x1, x2, ..., xi there is at least one burst state xp, where 1 p i. Suppose that the underling regular states of the burst state xp are x1 p, ..., xk p. Then the infinite cycle actually involves the following regular states: x1, ..., xp-1, xj p, xp+1 ..., xi, where 1 j k. Hence there is an infinite cycle that involves i regular states of the machine t conflicts with the assumption that the machine has no infinite cycles. This completes the proof. 2.3 Observer As depicted in Figure 1-1, we build an output feedback loop w ith a controller C, which is also an asynchronous machine. Specifically, this controller C is composed of two asynchronous machine: an observer B and a state-feedback control unit F (Figure 2-3).

PAGE 28

28 Figure 2-3. Control configurati on for the closed-loop system c Here, the observer B estimates the uncer tainty caused by crit ical races with the input/output information of and generates estimate state of to feed F. With the external input of the whole system and the estimate state of the control unit F generates a sequence of input to drive to match the model. We denote the controller C with (F, B). We use the observer in a way that is similar as it is used in other branches of control theory. Specifically, the observer here is an asynchronous input/state machine, which has two functions: a) check if the asynchronous machine has reached its next stable state; b) use the input/output information to estimate the current state of Let g = (A, Y, X, x0, sg, hg) be the generalized machine derived from Similarly as in Geng and Hammer (2005), we can build an observer that reproduces all st able and detectable transitions of the machine g The observer for g is an input/state machine B = (A Y, X, Z, z0, I) with two inputs: the input character u A of g and the output burst Y of g the state set Z is identical to the generalized state set X and the initial condition is identical to that of g i.e., z0 =x0. The recursion function A Y Z of B is u y v F B x C o n t r o l l e r C

PAGE 29

29 constructed as follows. First, using the generalized stable recursion function sg, define the function A {0, 1} Z by setting (z, u, a) sg z, u) if a = 1; z if a = 0. (2-2) Now, assume that the machine g is in a stable combination (x, ui-1), when the input character changes to ui, where (x, ui) is also a detectable pair. The change of the input character may give rise to a chain of transitions of g. Let k i be a step during this chain of transitions, let k be the burst of g from step i to step k, and let uk be the input character of g at step k. Since fundamental mode operatio n requires that the input charac ter be kept constant during a chain of transitions, we have uk =ui. Define (x, uk, k) s x, uk) if k = (x, uk); x otherwise. (2-3) Let zk be the state of the observe r B at the step k, while k be the output of B. The observer B is then an input/sta te machine defined by the recursion B zk+1 = (zk, uk, k) k =zk (2-4) The observer B is a stable state machine. To describe the operation of the observer, assume that the observer switched to the generalized state x immediately after g has reached the stable combination (x, ui-1). Let p i be the step at which the ch ain of transitions from (x, ui) to the next stable state x = sg(x, ui) terminates; then, p = (x, ui). As the pair (x, ui) is detectable, it follo ws by the definition of that the output of the observer B switches to the state x at the step p+1.

PAGE 30

30 We can now summarize the implications of our recent observations on the control configuration Figure 1-1. Fundamental mode operation requires the output of the controller C to remain constant while the system g is in transition. In order to fulfill this requirement, it must be possible for the controller C to detect the point at which g has completed its transition process. As discussed above, the out put of the observer B switche s to the state that represents the next generalized stable state of g immediately after g has reached that stat e; this signifies the end of the transition process and indi cates the most recent stable state of g. In this way, the observer B helps create an envi ronment in which the machine g can be controlled in fundamental mode operation.

PAGE 31

31 CHAPTER 3 REACHABILITY OF A GENERALIZED MACHINE The occurrence of critical ra ces in an input/output asynchronous machine causes the lacks of information about the exact state of the machin e. We use the concept of generalized states to deal with this uncertainty and keep a machine op erate in fundamental mode. In this chapter, we use generalized states to characterize the re achability properties of an asynchronous machine with critical races. First, let me introduce some important concepts that will be used in latter part of this chapter. 3.1 Generalized Reachability Matrix Definition 3-1 Let = (A, Y, X, x0, f, h) be an asynchronous machine with the state set X = {x1, xn} and let g = (A, Y, X, x0, sg, hg) be the generalized machine associated with where X = { x1, xn, xn+1, xn+t} is the generalized state set. The generalized one-step reachability matrix R g is defined as a (n+t) (n+t) matrix with entry Rij, where Rij is the set of all characters a A for which xj sg(xi, a) and for which the transition xi xj is a detectable transition. If there is no such character a, then set Rij N, where N is a character not in the alphabet A. Note that when the generalized machine g is equal to the machine (i.e., when there are no burst states), then the ge neralized one-step reachability ma trix reduces to the one-step reachability matrix R In view of the earlier disc ussion in Geng and Hammer (2005) only transitions that are both stable and detectable can be used when constructing a controller. The stability of the transition is guarantied by the ge neralized stable recursion func tion of the controlled machine

PAGE 32

32 However, the detectability of each transition needs to be checked according to proposition 2-10 in the construction of the genera lized one-step reachability matri x. Therefore, each entry of the generalized one-step reachability matr ix characterizes if the machine g can go from one generalized state to another through a stable and detectable transition. Let = (A, Y, X, x0, f, h) be an asynchronous m achine with the state set X = {x1, xn} and let g = (A, Y, X, x0, sg, hg) be the generalized m achine associated with where X = { x1, xn, xn+1, xn+t} is the generalized state set and Xb = { xn+1, xn+t } X is the burst state set. According to Definition 3-1, we can obtain the one-step reachability matrix R of the machine For the generalized machine g, the construction the generalized one-step reachability matrix R g contains two tasks: 1) Determine the necessary burst states; and 2) Add to the reachability matrix rows and columns corresponding to the burst states. Then, we can divide R( g) into 4 blocks R( g ) = R 11 | R 12 _ _R 21 | R 22, where, R11 is n n; R12 is n t, R21 is t n, and R22 is t t. The matrix R11 describes one-step deterministic transiti ons among regular states of while R22 describes one-step transitions among burst states. The submatrix R12 represents one-step transitions from regular states to burst states, while R21 represents one-step tr ansitions from burst stat es to regular states. Example 3-2 Consider the machine with transition table of Table 2-1, which has the input alphabet A = {a, b, c}, the output al phabet Y = {0, 1, 2}, and the state set X = {x1, x2, x3, x4, x5}. There is a critical race pair (x1, c) in the machine

PAGE 33

33 The generalized machine g derived from the has a generalized state set X = {x1, ..., x5}, and it can be depicted as follows. Table 3-1. Stable state transition table of the machine g x1 x5 x2 x4 b c c a c b c x3 a a a a b Figure 3-1. State flow diagram of the machine g According to Definition 3-1, the one-step reach ability matrix of the original machine is R( ) = accb acNb aNcN aNNb a b c Y x1 x1 x4 x5 0 x2 x1 x4 x2 1 x3 x1 x3 1 x4 x1 x4 2 x5 x1 x5 1

PAGE 34

34 and the generalized one-step reachab ility matrix of the machine g is R( g) = aNNbc acNbN aNcNN aNNbN aNNNc The submatrix R22 = [c] in the matrix R( g) describes the stable and detectable transitions inside the burst state set Xb. After obtaining the generalized stable recursion function sg we can get the generalized one-step reachability matrix R( g) as above. Similar to the one-step transition matrix in Venkatraman and Hammer (2004), we define so me operations on the generalized one-step reachability matrix R( g). Based on these operations we can obt ain the overall vi ew of reachable states in the generalized machine g and the information about how to approach the destination of any transition. The latter information is very us eful in the construction of the controller of the closed loop system. Definition 3-3 Let A be the set of all stri ngs of characters of the alphabet A and let wi be a subset of A or the character N, i = 1, 2. The operation of unison is defined by w1 w2 := w1 w2 if w1 w2 Aw1 if w1 A and w2 = N w2 if w1 = N and w2 AN if w1 = w2 = N. (3-1)

PAGE 35

35 The unison C : A B of two n n matrices A and B, whose entries are either subsets of A or the character N, is defined entrywise by Cij : Aij Bij, i, j = 1, ..., n. Note that N takes the role of zero. For example, given two 3 3 matrices A = abN bNN aac and B = NbN cNN abb the unison of A and B is C : A B abN {bc}NN a{ab}{bc} Definition 3-4 Let A be the set of all stri ngs of characters of the alphabet A and let w1, w2 be two subsets of A or the character N. C oncatenation of elements w1, w2 A N is defined by conc(w1,w2) := w2w1 if w1 w2 AN if w1 = N or w2 = N. (3-2) Let W = {w1, w2, wq} and V = {v1, v2, vr} be two subsets, whose elements are either subsets of A or the character N. Define conc(W, V) := i = 1 q j = 1 r conc(wi,vj) (3-3) For instance, consider two subsets W = {a,N,{bc}} and V = {N,a,{ab}}. The concatenation of W and V is conc(W, V) = {a,aa,{aa, ba},N,{a,b},{b,c},{ab,ac},{ab,bb,ac,bc}}. Definition 3-5 Let C and D be two n n matrices whose entries are either subsets of A or the character N. Let Cij and Dij be the (i,j) entries of th e corresponding matrices. Then, the product Z : CD is an n n matrix, whose (i,j) entries Zij is given by

PAGE 36

36 Z ij : n k=1 conc(Cik,Dkj), i,j = 1, ..., n. (3-4) For example, consider two 3 3 matrices A = abN bNN aac and B = NbN cNN abb Then the product of A and B is Z = AB = cbbaN NbbN {acca}{babc}bc Using the operation of product, we can de fine powers of the generalized one-step reachability matrix by setting Rq( g) : Rq-1( g)R( g), q = 2, 3, ... (3-5) Proposition 3-6 All transitions of the matrix Rq( g) are stable and detectable transitions. Proof According to the definition of the gene ralized one-step reachability matrix, each non-zero entry of the matrix refers to a stable and detectable transition. After the operation of product, every non-zero entry of a generalized multi-step reacha bility matrix refers to a combination of multi-step stable and detectable transitions. This operation does not change either the stability or the detectability of the transitions. Thus, for every entry of the matrix Rq( g), if it is not N, it stands for a stable and detectable transition. Based on Proposition 3-6, for an integer q 1, the matrix Rq( g) describe if the machine can reach one state from another st ate through exact q stable and detectable transitions. If the (i,j) entry of the matrix Rq( g) is not N, then it is the set of all input strings that can takes the machine g from the state xi to the state xj via a q-step stable and de tectable transition. If the (i,j) entry is N, then it is impossible to reach the state xj from the state xi in exact q stable and detectable transitions. Though, it might be possible to reach from xi to xj in stable and

PAGE 37

37 detectable transitions, where < q or > q. Since we are also interested in if the machine can reach one state from another in fewer transi tions, it is needed to construct a multi-step reachability matrix. Definition 3-7 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine and let R( g) be the generalized one-step reach ability matrix of the machine g. The generalized qstep reachability matrix is defined by R(q)( g) : r = 1, ..., qRr( g q = 2, 3, ... (3-6) Note that the (i,j) entry of R(q)( g) contains the reachability information from the state xi to the state xj. If the (i,j) entry is no t N, it consists all strings that may take the machine g from xi xj through stable and detectable transitions in q or fewer steps. It leads to the following statement and its proof is si milar to Murphy, Geng and Hammer (2003). Lemma 3-8 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine with n states and t burst states, and let R( g) be the generalized reachability matrix of the machine g. Then the following two statements are equivalent: (i) The generalized state xj is stably reachable through a detectable transition from the generalized state xi. (ii) The (i,j) entry of R(n+t)( g) is not N. Proof. Let A be the set of all strings of characters of the alphabet A. If the first statement is true, namely, the state xj is stably reachable from xi, then there is an input string u : uk-1 u1u0, which satisfies xj = sg(xi, u) and the transition from xi to xj is

PAGE 38

38 detectable Here, u A and k : |u |. If |u | (n t 1), then take u : u Thus, the (i,j) entry of R(n+t)( g) is uk-1 u1u0. If |u | (n t 1), then we need to show that a shorter string u in the string u still satisfies xj = sg(xi, u) and |u| (n t 1). Define recursivel y a string of states x0, x1, ..., xk, by setting x0 : xi and xm+1 : sg(xm, um), for m = 0, 1, ..., k-1. This implies xk = xj. The length of the string x0, ..., xk, k +1 is greater or equal to (n t). However, there are totally only (n t) distinct generalized states of the machine g. So, at least one generalized state must be repeated in the string x0, ..., xk. Suppose xp = xq, for 0 p q k. Remove from u the string v, which satisfies xq = sg(xp, v). Afterwards the s hortened input string u : u0u1 up-1uq uk-1 uk-1 uqup-1 u1u0 (or u : uk-1 uq when p = 0 ). This shortened u still satisfies xj = sg(xi, u ). Keep shortening the input string until an input string u of length |u| (n t 1) still satisfying xj = sg(xi, u). Conversely, if the second statemen t is true, then it implies that there is an input string u A of length |u| (n t 1). Suppose the (i,j) entry of R(n+t)( g) is the string u : uk-1 u1u0. There are k input characters in the string u and those characters satisfy the following equation: setting x0 : xi and xm+1 : sg(xm, um), for m = 0, 1, ..., k-1. Then we have xk = xj. According to the definitions of the generali zed stable recursion function, the machine g is only involved into stable transitions here. Meanwh ile, in the construction of the generalized reachability matrix R( g), all undetectable transitions are el iminated. Thus, the generalized (n t

PAGE 39

39 1) step reachability matrix R(n+t)( g) only contains detectable tr ansitions. So, the generalized state xj is stably reachable from the generalized state xi. Therefore, all possible stable and det ectable transitions for the machine g can be found in the matrix R(n+t)( g), i.e., the generalized (n t 1) step reachability matrix characterizes the reachability property of the machine g with n states an d t burst states. Definition 3-9 Let R( g) be the generalized one-step reachability matrix of the machine g, which has n t generalized states. The generalized st able reachability matr ix of the machine g is ( g) : R(n+t)( g). Example 3-10 Consider the machine and the generalized machine g of Example 3-2. g has a generalized state set X = {x1, x2, x3, x4, x5} and (n t 1) = 4. Raise the power of the R( g) as follows: R2( g) = {aabaabbbac}NNac{cacbcc} ba{ccca}N{acaa}N {aabaac}NccNca {abbbba}{ccca}N{acaa}cb {aabaac}caN{aaac}{cacc} After a stable transition, repeat applying the same input charac ter will not change the state of the machine. Thus, all same consecutive input character can be replace d by one character. For instance, the input string aa can be replaced by a and it will not affect the stable transitions of the machine. Hence, we obtain

PAGE 40

40 R2( g) = {abbaabac}NNac{c cacb} ba{cca}N{aac}N {abaac}NcNca {babba}{cca}N{aac}cb {abaac}caN{aac}{cca} Continue raising the power of the R( g) until the (n t 1) = 4. Then we have R4( g) = aabacabaaca abababacacabacac NN bbababbac bababaca ccacabcac cabacaca aabacabaabc acaabababac cN bbabcbabbac babababcbaca cacabcaccaba cabccaca aacabaaca abacacac N c{babcbacbacababa}{}cacaccabacaca aababaaca ababacab NN{}bbababbababaca {}cacabcabacaca aacabaaca abacacac NN {}babacbababaca{} ccacaccabacaca According to Definition 3-7 and Definiti on 3-9, we obtain the generalized stable reachability matrix g) of the machine g as follow:

PAGE 41

41 ( g ) = aabacabaaca abababacacabacac NN bbababbac bababaca ccacabcac cabacaca aabacabaabc acaabababac cN bbabcbabbac babababcbaca cacabcaccaba cabccaca aacabaaca abacacac N c{babcbacbacababa}{}cacaccabacaca aababaaca ababacab NN{}bbababbababaca {}cacabcabacaca aacabaaca abacacac NN {}babacbababaca{} ccacaccabacaca. 3.2 Common-output Generalized States As we have mentioned before, if the outcomes of a critical race (r, v) can be divided into more than one burst equivalent set, then sg(r, v) consists of more than one generalized state. This situation is shown in the genera lized one-step reachability matr ix as one input string appears more than once in different entries of a single row. Consider both this fact and the Lemma 3-8, we have the statement below. Proposition 3-11 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine with n states and t burst states, and let ( g) be the generalized stab le reachability matrix of the machine g. Then the following two statements are equivalent for all input strings u A+ and for all j = 1, ..., n t. (i) Applying u at the generalized state xi generates a critical race. (ii) The string u appears in more than one entry of row i of the matrix ( g). Note that the above conclusion is similar w ith the Proposition 4-16 in Venkatraman and Hammer (2004).

PAGE 42

42 After using burst states to repr esent subsets of states which have the same output value and same burst, we can see there are still critical ra ces in the machine on the generalized state base. Some input strings may be repeated in more than one entries of a row of the generalized stable reachability matrix (Example 3-10). That is caused by the existence of critical races. In the present discussion, the machine ( g) is an input/output machine. Thus, what matters to the user are the output value but not the state of the ma chine itself. Next we check if the machine can be led from different outcomes of th e critical races to the same output value. In Venkatraman and Hammer (2004), if the machine can be led from differ ent outcomes of the critical races to the same state, then it means the existence of a feedback trajectory. Here, we can loose the restriction to a subset of states which have the same output value. They can be also called common-output generalized states. Definition 3-12 Let g = (A, Y, X, x0, sg, hg) be a generalized asynchronous machine with a generalized state set X = Xb X. Assume that the machine g have critical races (r1, v1), (r2, v2), (r, v), and let T(ri, vi) := {r1 i, r2 i, rm(i) i} X be the set of all outcomes of the critical race (ri, vi), i =1,, Divide T(ri, vi) into subsets C1 i, C2 i, Cm(i) i according to the output value of r1 i, r2 i, rm(i) i. These subsets C1 i, C2 i, Cm(i) i can be represented by x1, x2, ..., xc, which are called the common-out put states of the machine g. Set m = n t c, then the generalized state set increases to X = {x1, ..., xm}. Note that the outcomes of the critical race s may be a combination of burst states and regular states. Moreover, if two subsets contain the same states, then they are represented by the same common-output state.

PAGE 43

43 After introducing the common-out put state of the machine g, we should update the generalized one-step reachability matrix R( g) and the generalized reachability matrix ( g). The transitions from one generalized state to a subset of states, which have the same output value, will be replaced by the single transi tion from the starting state to a newly defined common-output state. Example 3-13 Consider a machine with the input alphabet A = {a, b, c}, the output alphabet Y = {0, 1, 2}, and the state set X = {x1, x2, x3, x4, x5}. There is a critical race pair (x1, c) in the machine (Table 3-2). Table 3-2. Transition table of the machine a b c Y x1 x1 x4 {x4, x5} 0 x2 x2 x1 x2 1 x3 x1 x5 x3 1 x4 x1 x4 x2 2 x5 x2 x5 x3 2 The stable state machine of is s (Table 3-3). Table 3-3. Transition table of the machine s a b c Y x1 x1 x4 {x2, x3} 0 x2 x2 x4 x2 1 x3 x1 x5 x3 1 x4 x1 x4 x2 2 x5 x2 x5 x3 2 Using Algorithm 2-13, we get the genera lized stable recu rsion function sg of the machine g. Associate a burst state x6 with the subset {x2, x3}. Then the generalized machine g has a generalized state set X = {x1, ..., x6}. Assign a common-output state x7 to represent the subset

PAGE 44

44 {x4, x5}. Now the generalized machine g has a generalized state set X = {x1, ..., xm} and m 7. The generalized one-step reachability matrix is R( g) = aNNbNcN N {ac} N b N N N aNcNbNN acNbNNN NacNbNN aaNNNcb aaNNNcb Definition 3-14 Let |s be a generalized stable state machine with the generalized stable recursion function sg, and let u be an input string of |s. The transition induced by u from a generalized state x is a de terministic transition if sg(x, u) consists of a single state. The machine g is a deterministic machine if all transitions of |s are deterministic. 3.3 Output Feedback Trajectory From the definition of the critical race, we know a transition from a critical race pair to the outcomes is not a deterministic transition. So, or iginally, an asynchronous machine with critical races is not a deterministic machine. If we can transform the machine with critical races into a deterministic machine on a specific basis, then we actually get rid of the e ffect of the critical races to the machine. The following procedure helps us to transform a machin e with critical races into a deterministic machine. Consider a generalized machine g with the generalized state set {x1, x2, ..., xm} and input set A. Assume the m achine has a critical race (xj, v) with the outcomes {xp, xq} and p

PAGE 45

45 q. If the h(xp) = h(xq), then we can define a burst state or a common-output state to make the transition from xj to the subset {xp, xq} a deterministic transition. So, we only need to focus on the situation that the outcomes have different output values, i.e., h(xp) h(xq). In another word, we cannot make the generalized machine g transform into a deterministic one simply with a generalized state set. When h(xp) h(xq), if there exist input string s which can take the machine g from these two states (also two output values) to a single ge neralized state xs through deterministic transitions, then the effect of the critical race (xj, v) can be eliminated. Assume there exist input strings u1, u2 A, where u1 takes the machine from xp to xs deterministically and u2 takes the machine from xq to xs deterministically. That means sg(xp, u1) = sg(xq, u2) = xs. Then we can generate a deterministic transition from xj to xs with introducing an output feedback controller to the machine g as follows: After applying the input v at the state xj, check the outcomes. If the outcome is xp, then apply u1 to the machine. If the outcome is xq, then apply u2 to the machine. Hence, based on the generalized state set of a generalized machine g, we can turn the machine g into a deterministic machine with an output feedback controller. This co ntroller sets up a standard pr ojection on the generalized state set X, which can be denoted by x : X A X : x(x, u) = x. Definition 3-15 Let g be an asynchronous machine w ith the generalized state set X = {x1, ..., xm}, the input alphabet A, and the gene ralized stable recursion function sg. An output feedback trajectory from the generalized state xj to the common-output state xi is a list {S0, S1, ..., Sp} of sets of valid pairs of |s with the following properties: (i) sg (x, u) is a detectable transition for all (x,u) Uj=0,...,p Sj,

PAGE 46

46 (ii) S0 = {(xj, u0)}, (iii) sg [S] x[S], = 0, ..., p 1, (iv) sg [Sp] = {xi}. For example, consider the machine g in Example 3-13. The out put feedback trajectory from the generalized state x6 to the generalized state x4 is: S0 = {(x6, a)}, S1 ={(x1, b), (x2, b)}. Proposition 3-16 Let g be a generalized machine and let xj and xi be two generalized states of g. The following two statements are equivalent. (a) There exists an output feedback controller C that drives g through a deterministic transition from xj to xi. (b) There is an output feedback trajectory from xj to xi. Proof. Suppose that the part (b) is valid, and let { S0, S1, Sp} be an output feedback trajectory from xj to xi. We construct an output feedback cont roller C that takes the machine g from xj to xi through a string of deterministic transiti ons. This output feedba ck controller C has two inputs: one is the output burst Y of g, and the other is the external input v A, which is also the command input of the cont roller C. Given a set of characters W A, we need to construct a controller C(xj, xi, W). This controller takes the machine g from generalized state xj to xi through deterministic transitions as th e response to an input character w W. The controller C(xj, xi, W) is a combination of an obser ver B and a control unit F as shown in Figure 2-3. Where, the observer is an input/state machine B = (A Y, X, Z, z0, I) with two inputs: the input character u A of g and the output burst Y of g the state set Z is identical to the generalized state set X and the initial condition is identical to that of

PAGE 47

47 g i.e., z0 =x0. And the function A Y Z is the stable recurs ion function of B. Let zk be the state of the observer B at the step k, while k be the output of B. The observer B is an input/state machine defined by the recursion (Eq. 2-4) The control unit F is also an i nput/state asynchronous machine F = (A X, A, 0, ) with two inputs: th e external input v A and the output X of the observer B. To complete the construction of the controller C, we need to derive the recursion function and the output function of the unit F. According to Figure 2-3, as long as v W, the controller C stays in its initial state (z0, 0) and the input character u of the machine g equals to the external input character v. After the machine g arrives a stable combination of state xj, if the external input v changes to a character of W, then the cont roller C starts working. The obse rver B collects both the input character u and the output burst xj of the machine g and feeds the contro l unit F with the state of the machine g. The control unit F generates a string of characters u1u2ur and apply it to the machine g. This input string u1u2ur will drive the machine g from the state xj to xi through a string of detectable and stable deterministic transitions. Recalling that the control unit is an i nput/state asynchronous machine F = (A X, A, 0, ). The recursion function of F is a function X and the output function of F is denoted by X A. Referring to Figure 2-3, the output X of the observer B is one of the inputs of F, and the ot her input is the external input v W. Then the control unit F generate the string u A to feed the controlled machine g. Note that the cont rol unit F must

PAGE 48

48 operates in a fundamental mode, so the whole sy stem must have reached a stable combination before the F generates the next input character for g. Assume that F will generate r input characters u1u2...ur to feed g, then it needs r states 1(xj, w), 2(xj, w), ..., r(xj, w). Denote this set by xj, w : { 1(xj, w), 2(xj, w), ..., r(xj, w)}. We define the recursion function and output function of the F as follows. (i) Let U(xj) A be the set of all input characters that form stable combination with the generalized state xj, and let z0 be the initial state of B and 0 be the initial state of F. Set ( 0, (z, t)) : 0 for all (z, t) X A\xj U(xj), ( 0, (xj, v)) : 1(xj) for all v U(xj). Where 1(xj) is the state of F, when observer B detects a stable combination with xj. When both B and F are at initial states, the controller C(xj, xi, W) directly applies the external input v to g, thus set ( 0, (z, v)) : v for all (z, v) X A. (ii) When the observer B det ects a stable combination of g with the generalized state xj, suppose the external input switches to a character w W. We choose a character uj U(xj) and set ( ( xj), (xj, t)) : uj for all t U(xj). In this way, the machine g lingers in the state xj when the external input switches to a character of W. Hence, the fundamental mode operation of the machine is guaranteed. Then the control unit F will generate an input string u=u1u2...ur to drive the machine g to the

PAGE 49

49 generalized state xi. Since we have a output feedback trajectory {S0, S1, ..., Sp}, we need P new states for F, where P = # xS0 + # xS1 # xSp Denote the state of F as k(xj, w, x), where x xSk and k = 1, ..., p. When the input character switches to w, the c ontrol unit F moves to the state 0(xj, w, xj) and it begins to generate the first input character u0 to feed g, where u0 A is an character that satisfies (xj, u0) S0. To implement this, we set ( 1(xj), xj, w) : 0(xj, w, xj) for all w W; ( 1(xj), xj, v) : 1(xj) for all v U(xj) \ W; ( 1(xj), xj, v) : 0 for all v U(xj) W; ( 0(xj, w, xj), x, v) : u0 for all (x, v) X A. After u0 is applied to g, the machine will move to a gene ralized stable combination with a generalized state x1xS1. After the observer B detects th is transition, the control unit F moves to the next state 1(xj, w, x1) and generates the ne xt input character u1 to the machine g. The process continues similarly until xi is reached. Then at the step k {1, 2, ..., p} the function and must be defined as follows ( k-1(xj, w, xk-1), xk, w) : k(xj, w, xk), ( k(xj, w, xk), x, v) : uk for all (x, v) X A. At the k=p step, the machine g reaches a state xpxSp. Set sg(xp, up) : xi, where up is an character satisfies (xp, up) Sp. This is accomp lished by setting ( p(xj, w, xi), xk, w) : p+1(xj, w, xi),

PAGE 50

50 ( p+1(xj, w, xi), z, t) : for all (z, t) X A\W. ( p+1(xj, w, xi), x, v) : up for all (x, v) X A. As long as the external input remains as a ch aracter in set W, th e machine will linger in the stable combination (xi, up). If the external input is no long er belongs to W, the controller returns to the initial state We build the controller as well as prove that statement (b) implies statement (a) as above. Conversely, assume part (a) is valid. Let 0 be the initial state of the controller C and let C( x, u) be the output value produ ced by the controller C when it is at the next stable state corresponding to its state g is at the state x and the ex ternal input is u. By assumption, there is an external input value w that induces the controller C to ge nerate an input string u0u1...up for the machine g from the generalized state xj to the generalized state xi via deterministic transition. The first char acter of this input string is u0 = C( xj, u). Define the set S0 : {(xj, u0)}. Let sg be the generalized stable recursion function of g. when the input from the controller to g changes to u0, g moves to a generalized stable combination with one of the states of the set s(xj, u0) s[S0]. When this state is reached by g the controller C detects the new state and controller moves to its own next stable state. Let (x, u0) be the next stable state of the controlle r and let u1 C( (x, u0), x, w) A be the output character generated by the controller once C reaches (x, u0). Define the set S1 : {(x, C( (x, u0), x, w)) : x s(S0)}. Continue operating like that until the set Sk, k > 0, is defined. Bu ild a new set by setting

PAGE 51

51 Sk+1 : {(x C( (x uk), x w)) : x s(Sk)}. By assumption, the controller C drives g to the state xi through deterministic transitions. Consequently, there exists an integer p such that s(Sp) xi. Then, the list S0, S1, ..., Sp forms a output feedback trajectory. So, th e existence of a output feedback controller C that drive g from xj to xi through deterministic transitions, implie s the existence of a output feedback trajectory S0, S1, ..., Sp from xj to xi. Namely, part (a) implies part (b). This completes the proof. 3.4 Preliminary Generalized Skeleton Matrix Using the algorithm in the above proof of Proposition 3-16, we can check the basic connection between any two genera lized states of the machine g, namely, the existence of the output feedback trajectory between any pair of generalized states. If we focus on the stable and detectable reachability properties, then we dont n eed to record all the input strings. Instead, we can use a numerical matrix, which has only entrie s of one and zero, to re present this one-step stable and detectable reachabil ity properties. This numerical matrix can be called preliminary generalized skeleton matrix of the machine g. Afterwards, it is easier to calculate the power of it and obtain the overall preliminary genera lized skeleton matrix. For the machine g with m generalized states, the overall preliminary genera lized skeleton matrix characters all the stable and detectable transitions among the generalized stat es of the machine within m steps. We can use the following algorithm to gradually transfor m the generalized one-step reachability matrix into the preliminary one-step generalized sk eleton matrix. Meanwhile, the machine g is transformed into a deterministic machine on the ge neralized state basis with an output feedback controller C.

PAGE 52

52 An operation involving strings of A+ and zero and 1 should be defined before giving the algorithm. Definition 3-17 Let be a character not included in A. The meet operation between two strings of A+ and zero and 1 is defined as follow: 0 0 : 0, 0 1 1 0 : 0, 1 1 : 1, 0 a a 0 : 0, 1 a a 1 : for all a A+. The meet of two vectors with r 1 components is defined entr ywise as the vector of the meets of the corresponding components. Algorithm 3-18 Let g be an asynchronous machine w ith the generalized state set X = {x1, ..., xm} and let ( g) be the generalized stable reachability matrix of the generalized machine g. Step 1: Transpose the matrix ( g) and denote the resulting matrix by ( g). Step 2: Replace all entries of N in the matrix ( g) by the number 0; denote the resulting matrix by K1. Step 3: Perform (a) below for each i,j = 1, ..., m; then continue to (b): (a) If K1 ij includes a string of A+ that does not appear in any other entry of the same column j, then replace entry K1 ij by the number 1. Otherwise, let the entry K1 ij remain. (b) Set k : 1 and denote the resulting matrix by K(k). Step 4: If all entries of row k of the matrix K(k) are 1 or 0, then set K(k+1) : K(k) and set k : k+1. Step 5: If k = m + 1, then set K( g : K(k) and terminate the algorithm. Otherwise, go to step 6.

PAGE 53

53 Step 6: Perform the following operations: (a)If there is a character u A that appears in row k of K(k), then let j1, ..., jq be the columns of row k of K(k) that includ e u. Denote by J(u) the meet of rows j1, ..., jq of the matrix K(k). (b) If J(u) has no entries other than 0 or 1, then delete u from all entries of row k of the matrix K(k); set all em pty entries, if any, to the value 0. Continue to (c). (c) If J(u) has no entries of 1, then retu rn to Step 3. Otherwise, continue to (d). (d) If J(u) has entries of 1, then let j1, ..., jr be the entries of J(u) having the value 1. Let S(k) be the set of rows of K(k) that consists of row k and of every row that has the number 1 in row k. In the matrix K(k), perform the following opera tions on every row of S(k): Delete from the column all occurrences of input characters that appear in columns j1, j2, ..., jr of the row. Replace rows j1, j2, ..., jr of the column by the number 1. If any entries of K(k) remain empty, then replace them by the number 0. Return to Step 4. The final resulting matrix K1( g is called the preliminary gene ralized skeleton matrix of the machine g. Definition 3-19 The outcome K1( g) of Algorithm 3.23 is defined as the preliminary generalized skeleton matrix of th e generalized asynchronous machine g.

PAGE 54

54 Note that this preliminary generalized skeleton matrix K1( g) of the generalized machine g is similar to the one-step fused skeleton matrix ( ) of an asynchronous machine in Geng and Hammer (2005). Example 3-20 Consider the machine and the generalized machine g of Example 310. The generalized stable reachability matrix g) of the machine g is ( g ) = aabacabaaca abababacacabacac NN bbababbac bababaca ccacabcac cabacaca aabacabaabc acaabababac cN bbabcbabbac babababcbaca cacabcaccaba cabccaca aacabaaca abacacac N c{babcbacbacababa}{}cacaccabacaca aababaaca ababacab NN{}bbababbababaca {}cacabcabacaca aacabaaca abacacac NN {}babacbababaca{} ccacaccabacaca. Applying the Algorithm 3-18 to the matrix ( g), we can obtain the preliminary generalized skeleton matrix K1( g of the machine g is K1( g) = 10011 11011 10111 10011 10011

PAGE 55

55 Proposition 3-21 Let g be a generalized machine w ith the preliminary generalized skeleton matrix K1( g and let xi and xj be two generalized states of g. Then the following two statements are equivalent. (a) There exists an output feedback trajectory from xj to xi. (b) The (i, j) entry of K1( g is 1.

PAGE 56

56 CHAPTER 4 MODEL MATCHING FOR INPUT/OUTPUT ASYNCHRONOUS MACHINES WITH RACES In the present chapter we start to address the model-matching problem for input/output asynchronous machines with critical races. In la st chapter the reachabil ity properties of an input/output asynchronous machine with critical races has been discussed and corresponding generalized machine has been derived. The newl y defined generalized machine with related generalized state set and genera lized functions of the machin e could be controlled as a deterministic machine without critical races. Th e control of this kind of asynchronous machines has been discussed in Geng and Hammer (2005). T hus, the controller will be designed to correct the input/output machine under th e configuration of Fig. 2-3, so that the closed-loop system possesses an equivalent input/output behavi or as that of a prescribed model. Since we are discussing the input/output mach ines, we first study the equivalent list of the generalized machine g, with respect to the model Then we work on the sufficient and necessary conditions of the existe nce of the output feedback contro llers so as to solve the model matching problem. When such a controller exis ts, we provide an algorithm to construct the controller. Finally, an example is presented to illustrate how the control system operates. 4.1 Model Matching Problem As we mentioned before, the desi gn of a controller to eliminat e the effects of critical races of an existing asynchronous machine is calle d the Model-Matching Pr oblem. Specifically, the formal statement of the model matc hing problem is as follows. Let be a machine that exhibits undesirable behavior. Assume that the desirabl e behavior is specifi ed by an asynchronous machine The machine is called the model. Our objective is to design a controller C for which the behavior of the closed loop system c simulates the behavior of the model It is

PAGE 57

57 indicated in Kohavi (1970) th at the practical performance of an asynchronous machine is determined by its stable-state behavior. Thus, the stable-state behavior of c need to be equivalent to the stable-state behavior of Let us first introduce th e classical notions of equivalence. Definition 4-1 Let = (A, Y, X, x0, f, h) and = A, Y, X f h be two machines having the same input and the same output sets, and let |s and |s be the stable state machines induced by and respectively. Two states x X and X are stably equivalent (x ) if the following conditions are true: When |s starts from the state x and |s starts from the state then (i) |s and |s have the same permissi ble input strings; (ii) |s and |s generate the same output string for every permi ssible input string. The two machines and are stably equivalent if their initial stat es are stably equivalent, i.e., if x0 Note that two machines = (A, Y, X, x0, f, h) and = A, Y, X f h that are stably equivalent appear identical to a user. Definition 4-2 Given a machine and find necessary and sufficient conditions for the existence of a controller C such that c is stably equivalent to and operates in fundamental mode. If such a controller C exis ts, derive an algorithm for its design. In this dissertation, the model matching probl em concentrates on matching the stable input/output behavior of the model. The model can be taken as a st ably minimal machine. Let g be a generalized machine with the generalized state set {x1, x2, ..., xm} which is induced from the machine Our objective is then to match the i nput/output behavior of the generalized machine g and the model

PAGE 58

58 Next, let us introduce a notion which underlies the solution of the model matching problem for asynchronous machines. Given two sets S1 and S2 and a function g : S1 S2, denote by gI the inverse set function of g; i.e., for an element s S2, the value gI (s) is the set of all elements S1 that satisfies g( ) = s. Definition 4-3 Let = (A, Y, X, s, h) and = A, Y, X f h be two machines having the same input and the same output sets. Let g = (A, Y, X, sg, hg) be a generalized machine induced from the machine The state set X of consist of the q state 1, ..., q Define the subsets Ei : hg Ih ( i) X, i = 1, ..., q. Then, E( g, ) : {E1, ..., Eq} is the output equivalence list of g with respect to An equivalence list is characterized by th e following property: the value of the output function hg of g at any state of the set Ei is equal to the value of the output function h of at the state i. The members of an output equivalence list are not necessarily disjoint sets. Definition 4-4 Let g be a generalized machine with generalized state set X = {x1, ..., xm}, and let and be two nonempty subsets of X. The reachability indicator r( g, ) is defined as 1 if every element of can reach an element of through a chain of stable and detectable transitions; otherwise, r( g, ) = 0. Example 4-5 Let g be a generalized machine with generalized state set X = {x1, x2, x3} and the preliminary generalized skeleton matrix is K1( g

PAGE 59

59 K1( g = 111 111 001 Let = {x1, x2} and = {x2, x3} be two state subsets. Then r( g, ) = 1. Definition 4-6 Let g be a generalized machine with generalized state set X = {x1, ..., xm}, and let ..., q be a list of m 1 nonempty subsets of X. The fused skeleton matrix ( g, ) of is an q q matrix whose (i,j) entry is ij( g, ) = r( g, i, j). Example 4-7 Consider the machine g and the two state subsets and in the Example 4-5. Let be a list of subsets of X. Then the fused skeleton matrix ( g, ) of is ( g, ). ( g, ) = 11 01 Definition 4-8 Let ..., q and W={W1, ..., Wq} be two lists of subsets of X. The length of the list is the number q of its members. Th e list W is a subor dinate list of the list denoted as W if it has the same length q as the list and if Wi i for all i=1, ..., m. A list is deficient if it includes the empty set as one of its members.

PAGE 60

60 4.2 Existence of Controllers Next, we give the condition of the exis tence of a controlle r C for which c is stably equivalent to a specified model Given two p q numerical matrices A and B, the expression A B indicates that every entry of the ma trix A is not less than the corresponding entry of the matrix B, i.e., Aij Bij for all i = 1,..., p a nd for all j = 1, ..., q. Lemma 4-9 Let g=(A, Y, X, x0, sg, hg) and =(A, Y, X s h ) be asynchronous machines, where is stably minimal. Let X = { 1, ..., q} be the state set of where the initial condition of is 0 = d. Assume that there is a controller C for which c is stably equivalent to and operates in fundamental mode. The n, there is a non-def icient subordinate list of the output equivalence list E( g, ) for which ( g, ) K( ) and x0 d. The proof of the above Lemma 4-9 is similar to the proof of the Lemma 4.11 in Geng and Hammer (2005). The difference is that a generalized machine appears here instead of a regular machine Recall that all the underlying states of a burst state or a common output state in the generalized machine g are the same states in the original machine and those underlying states have the same output value. Thus, when applying a real input character u to the generalized machine g at a generalized state x it is the same to apply this input character u to the real machine at any underlying state of that generalized state x The real machine will generates the same output value as the generalized machine g does. The condition of Lemma 4-9 is not only a necessary condition, but also a sufficient condition for the existence of a controller to so lve the model matching problem. The inequality ( g, ) K( ) guarantees that the corr esponding output values of the two machines match. If the model has a stable transition from a state i to state j, then the machine g has a

PAGE 61

61 stable and detectable trans ition from every state in i to a state in j. Thus, we only need to construct a controller C which ge nerates the input string that takes g from a state in i to a state in j. This controller should be a combinati on of an observer a nd a control unit as described in Figure 2-3. Theorem 4-10 Let g=(A, Y, X, x0, sg, hg) and =(A, Y, X s h ) be stably reachable asynchronous machines, where is stably minimal. Let X = { 1, ..., q} be the state set of where the initial condition of is 0 = d. Then the following two statements are equivalent. (i) There is a controller C for which c = where c operates in fundamental mode and is well posed. (ii) There is a non-defici ent subordinate list of the output equivalence list E( g, ) such that ( g, ) K( ) and x0 d. Moreover, when (ii) holds, th e controller C can be desi gned as a combination of an observer B and a control unit F as depicted in Figure 2-3 and the obser ver is given by Equation 2-4. Proof The generalized machine g has a generalized state set X = {x1, x2, ..., xt}. All the underlying states of the generalized states {x1, x2, ..., xt} in this set X are the same states in the set X of the original machine Since all the real states incl uded in a burst state or in a common output state will work with the same input value, the same input value can be used on the real machine. Furthermore, the output of a bu rst state or a common output state is the same as that of the underlying states. Hen ce, the operation of the real machine is as same as before the

PAGE 62

62 introducing of the generalized machine. Thus, we can use the same method in Geng and Hammer (2005) on the generalized machine, i.e., to find a controller for g. The Lemma 4-9 indicates that statement (i) impl ies statement (ii). Now let us assume that (ii) is valid. Let ={ ..., q} be a subordinate list of E( g, ) satisfying ( g, ) K( ) and x0 d. Using we build a controller C for which the closed loop system c of 1.1 is stably equivalent to the model is well posed, and operates in fundamental mode. The controller C is a combination of an observer B and a control unit F as depicted in Figure 2-3. The observer B is given by Proposition 3-16, so we complete the proof by constructing the control unit F. Recall that the control unit F is an asynchronous machine F = (A X, A, 0, ) with two inputs: the external input v A and the output X of the observer B. To complete the construction of the controller C, we need to derive the recursion function and the output function of the unit F. Assume that is at the stable state i and that g is at a stable state i. Note that i is either the initial condition x0d of or the outcome of a dete ctable stable transition; is either the ini tial condition x0d of g or the outcome of a detectable stable transition. Assume the external input character switche s to the character w. Then the model moves to its next stable state s ( i, w)= j. Recall that sg is the generalized stab le recursion function of g. The inequality ( g, ) K( ) implies that there is an input string u=u1u2...ur such that the stable combinations ( u1), (sg( u1), u2), (sg( u1u2 ur-1), ur) are all detectable, and such that the state xr : sg( u) belongs to j. Define the intermediate states X1 : sg( u1), x2 : sg(x1, u2), xr = sg(xr-1, ur). (4-1)

PAGE 63

63 As the combinations (xi, ui), i= 1, r, are all stable and de tectable combinations, the states x1, xr appear as output values of the obs erver B immediately after having been reached by g. The situation can be depicted as follows. : i j g: i xr j Figure 4-1. Equivalence of two asynchronous machines and g The objective of the control unit F is to generate the string u = u1u2 ur and apply it as input to the real machine This action achieves model matchi ng for the present transition for the following reason. The string u drives the system g to the stable state xr, which then becomes the next stable state of the closed loop system c. Then, since h(xr) = h[ j] = h ( j), the next stable state of c produces the same output value as the model to match the models response. Note that the control unit F must operates in a fundamental mode, so the whole system must have reached a stable combination before th e F generates the next input character for g. Then, we construct a recursion function for F to implement the above behavior. Keeping in mind the requirement of fundamental mode operation of the machines, we need to make sure that the control unit F generates the string u one character at a time and at each step that the composite system has reached a stable combinati on before generating the next character. As the string u has r characters, the control unit F needs r states to accomplish this: 1( i, w), r( i, w). The resulting set of states w u1u2 ur

PAGE 64

64 i, w) : { 1( i, w), r( i, w)} is associated with the state i of the state of and the external input character w. To account for all possible such combinations, the control unit F needs the state set : 0 { i-1, i w i, }, where 0 is the initial state of F. We shall use the following notation. For a state x of the machine g, let U(x) : {a A : sg(x, a) = x} be the set of all input characters that form stable combinations with x. Similarly, for a state of the machine let U ( ) : {a A : s ( a) = } be the set of all input characters that form stable combinations with Recalling that the control unit is an i nput/state asynchronous machine F = (A X, A, 0, ). The recursion function of F is a function X and the output function of F is denoted by X A. Referring to the confi guration (2.24), the output X of the observer B is one of the inputs of F, a nd the other input is th e external input v W. Then, and are defined as follows. (i) Let the closed loop system c be at a stable combination, where g is at the state namely, the real machine is at one of the underlying st ates of the generalized state The observer B has he output value = and control unit F is at a state Select an element c U( and define ( ,( ,b)) : for all b U ( i),

PAGE 65

65 ( ,( ,a)) : c for all a A. This guaranties that the closed loop system c and the model operate in fundamental mode. (ii) Suppose that the external input switc hes to a character w satisfying s ( i, w) = j. Then, the control unit F needs to generate the input string u = u1u2 ur, to take g through the chain of states x1, xr to the state xr j. Meanwhile the output of the observer B will track the state sequence x1, xr. Thus, the recursion function is defined as follows. w)) : 1( i, w), k( i, w), xk, w)) : k+1( i, w), k = 1, 2, r-1, k( i, w), z, b)) : uk, for any (z,b) X k = 1, 2, r. (iii) In response to the last input character ur produced by F, the machine g reaches the desired stable state xr, which implies the real machine reaches one of the underlying states of the generalized stable state xr. The machine g needs to remain at the state xr until the external input switches from w to another character. Then, choose an element v U(xr) and assign r( i, w), xr, w)) : r( i, w), r( i, w), z, b)) : v, for all (z,b) X This completes the construction of the control unit F. Note that whenever the machine g is at a generalized stat e x, the real machine is at an corresponding underlying state x of this generalized state x and hg(x) = h(x ). This construction achieves model matching of the generalized machine g to the model with fundamental model operation as well. This concludes the proof.

PAGE 66

66 The proof of Theorem 4-10 includes an algorith m for the construction of a controller C solving the model matching problem. Then, we use the Algorithm 4.14 in Geng and Hammer (2005) to build a list that satisfies condition (ii) of th is theorem whenever such a list exists. This algorithm and Theorem 4-10 give a comprehe nsive and constructive solution of the model matching problem. A recursive process is used in the algorithm to build a decreasing chain of subordinate lists. The last list in this chain, if not deficient, sati sfies condition (ii) of Theorem 410; if the last list of the chain is deficient, then there is no controller that solves the requisite model matching problem. Let g=(A, Y, X, x0, sg, hg) and =(A, Y, X s h ) be the machines of Theorem 4.8, let E( g, ) = {E1, ..., Eq} be their output equiva lence list, and let K( ) be the skeleton matrix of The following steps yield a decreasing chain (0) (1) ... (r) of subordinate lists of E( g, ). The members of the list (i) are denoted by 1(i), ..., q(i); they are subsets of the state set X of g. Algorithm 4-11 Let g=(A, Y, X, x0, sg, hg) and =(A, Y, X s h ) be the machines of Theorem 4-10, let E( g, ) = {E1, ..., Eq} be their output equi valence list, and let K( ) be the skeleton matrix of The following steps yield a decreasing chain (0) (1) ... (r) of subordinate lists of E( g, ). The members of the list (i) are denoted by 1(i), ..., q(i); they are subsets of the state set X of g. Start Step: Set (0) := E( g, ). Recursion Step: Assume th at a subordinate list (k) = { 1(k), ..., q(k)} of E( g, ) has been constructed for some integer k 0. For each pair of integers i,j {1, ..., q}, let Sij(k) be

PAGE 67

67 the set of all states x i(k) for which the (i,j) element of ( g, j(k)) is 0; i.e., Sij(k) consists of all states x i(k) for which there is no chain of stable and detectable transi tions to a state of j(k). Note that Sij(k) may be empty. Then set T ij (k):= S ij (k) if K ij ( )=1; if K ij ( )=0. Now, using \ to denote set difference, define the subsets Vi(k) := j=1,...,qTij(k), i = 1, ..., q i(k+1) := i(k)\Vi(k), i = 1, ..., q Then, the next subordinate list in our decreasing chain is given by (k+1) := { 1(k+1), ..., q(k+1)}. Test Step: the algorithm terminates if the list (k+1) is deficient or if (k+1) = (k); otherwise, repeat the Recursi on Step, replacing k by k+1. 4.3 A Comprehensive Exampl e of Controller Design Consider an asynchronous machine = (A, Y, X, x0, f, h) with the input alphabet A={a, b, c}, the output alphabet Y={0, 1, 2}, and the state set X={x1, x2, x3, x4}. There is a critical race pair (x1, c) in the machine (Tables 2-1, Figure 2-1) Let another machine = (A, Y, X 0, f h ) be the desired model (Table 4-1, Figure 4-2). The initial state of is x0 = x1 and the initial state of is 0 = 1. After introducing a burst state x5 = {x2, x3}, we have the generalized machine g of able 4-2).

PAGE 68

68 Table 4-1. Transition table of the machine 3 c 1 2 b a b a a c Figure 4-2. State flow diagram of the machine Table 4-2. Stable state transition table of the machine g a b c Y x1 x1 x4 x5 0 x2 x1 x4 x2 1 x3 x1 x3 1 x4 x1 x4 2 x5 x1 x5 1 The initial state of g is x0 = x1. From Table 2-1 and Table 42., the output equivalence list is E( g, ) ={E1, E2, E3}, where E1={x1}, E2={x2, x3, x5}, E3={x4}. The preliminary generalized skeleton matrix of the generalized machine is K1( g) and the skeleton matrix of the model is K( ). a b c Y 2 1 2 1 3 3 3 2

PAGE 69

69 K1( g) = 10011 11011 10111 10011 10011 K( ) = 111 111 001 The subordinate list of the output equivalence list E( g, ) is x1 x2, x3, x5 x4 The fused skeleton matrix is ( g, ) that satisfies ( g, ) = 111 111 111 111 111 001 = K( ) Thus, there exists a controller C to turn the machine into a deterministic machine that matches the model We have a generalized machine g = (A, Y, X, x0, sg, hg) with the state set X ={x1, x5} and a subordinate list that satisfies ( g, ) K( ) and x1 According to the process of c onstruction of the controller C that is described before, we derive the control unit F and combine it with the observer B of Equation 2-4. Then, we have the corrective controller C as shown in Figure 2-3.

PAGE 70

70 Recall that the initial state of g is x0 = x1 and the initial state of is 0 = 1. Maintain the external input character as a to keep at the state 1 and maintain the external input character as a to keep g at the state x1. Now we construct F as follows. Set the initial state of F to 0. Denote the states of the observer B by {x1, x2, x3, x4, x5} and set the initial state of B to x1. Thus, we have U(x1) = {a} and U ( 1) = {a} and F: ( 0, (x1, a)) : 0, ( 0, (x1, a)) : a. B: x1, (a, x1 for all Y. Assume then the external input character sw itches from a to b. For the machine s ( 1, b) = 2 and h ( 1) = 0 and h ( 2) = 1, so this transition is detect able. In order to simulate this transition, the system g has to move to a state in 2 = x2, x3, x5 Since sg(x1, c)= x5. For the real machine it needs to move to either state x2 or state x3 and it does not matter in which state the really stays. In either case the control unit F needs to generate the character c to serve as input for so that F: ( 0, (x1, b)) (x1, 1, b), ( (x1, 1, b), (x1, b)) c. B: ( x1, (c, ) x 5 for = 01; x 1 otherwise. F: ( (x1, 1, b), (x5, b)) (x1, 1, b), ( (x1, 1, b), (x5, b)) c. B: x5, (c, x5 for all Y.

PAGE 71

71 Now consider the other option: the machine g is at a stable combination with the state x1 1 and is at a stable combination with the state 1, when the external input character switches from a to c. For the machine s ( 1, c) = 3 and h ( 1) = 0 and h ( 3) = 2. Thus this transition is detectable as well. To simulate this transition, the machine g needs to move to a state in 3={x4}, i.e. to x4 So does the real machine Since s(x1, b) = x4, the control unit F needs to generate the character b and this leads to the following F: ( 0, (x1, c)) (x1, 1, c), ( (x1, 1, c), (x1, c)) b. g: s(x1, b) = x4, (x1, b) = h(x1)h(x4) = 02 B: (x1, (b, ) x 4 for = 02; x 1 otherwise. Then, assume the machine stays at a stable combination with the state x 2 and the model is at a stable comb ination with the state 2, when the external input character switches to a. The models response is s ( 2, a) = 1 so F needs to generate an input character a to drive to a state in 1 ={x1}. So we have F: ( (x1, 1, b), (x2, a)) (x2, 2, a), ( (x2, 2, a), (x2, a)) a. g: s(x2, a) = x1, (x2, a) = h(x2)h(x1) = 10 B: (x2, (a, ) x 1 for = 10; x 2 otherwise.

PAGE 72

72 Another possibility is that is in a stable combination with the state x 2 and the model is at a stable comb ination with the state 2, when the external input character switches from b to a. The models response is s ( 2, a) = 1 so F needs to generate an input character a to drive to a state in 1 ={x1}. Then F: ( (x1, 1, b), (x3, a)) (x3, 2, a), ( (x3, 2, a), (x3, a)) a. g: s(x3, a) = x1, (x3, a) = h(x3)h(x1) = 10 B: (x3, (a, ) x1 for = 10; x3 otherwise. Assume further that g is in a stable combination with the state x 2 and the model is at a stable combination with the state 2, when the external input char acter switches from b to a. This case is a combination of the above two cases. Thus, F will ge nerate an input a to drive to a state in 1 ={x1}. F: ( (x1, 1, b), (x5, a)) (x5, 2, a), ( (x5, 2, a), (x5, a)) a. g: s(x5, a) = x1, (x5, a) = h(x3)h(x1) = 10 B: (x5, (a, ) x1 for = 10; x5 otherwise. This completes the construction of the corr ective controller C for the model matching problem. The state set of the control unit F is = { 0, (x1, 1, b), (x1, 1, c), (x2, 2, a), (x3, 2, a), (x5, 2, a)}.

PAGE 73

73 The above state set can be reduced to three stat es and F can be depi cted by Figure 4-3 (the notation near the arrows indicates the informati on of (output of the observer B, input of c)/output of F). 2 1 0 (x1, a)/a (x1, c)/b (x5, a)/a (x3, a)/a (x2, a)/a (x1, b)/c Figure 4-3. State transitions diagram of control unit F The observer B = (A Y, X, Z, z0, I) is defined according to Equation 2-4 and described by Figure 4-4. The controller is the comb ination of F and B according to Figure 2-3. x3 x2 x1 x4 (a,0) x5 (c,1) (b,2) (c,1) (c,1) (a,120) (a,120) (b,102) (c,01) (b,02) (a,20) (a,120) Figure 4-4. State transitions diagram of observer B

PAGE 74

74 CHAPTER 5 SUMMARY AND FUTURE WORK In the present work, the feedback controllers are introduced to correct the faulty behavior of asynchronous machines. When critical races af flict the asynchronous machine, the existence of the controllers offer a solution to eliminate th e effects of the critical races while controlling the machine to match a desirable race-free m odel. We call the problems as Model-Matching Problems. This approach disclose s an interesting and constructive field in which many related topics are worth investigating. The solutions have been obtained to th e Model-Matching Problem for asynchronous input/output machines. The concept of generalized state has been used to describe a persistent state of the machine about which only partial information is available. The generalized state allows us to use the partial inform ation available about the state of to continue controlling the machine as best as possible toward the goal of achieving model matching, while taking best advantage of the available information about The results of the Model-Matching Problem include necessary and sufficient c onditions for the existence of the controller, and algorithms for its construction whenever a controller exists. The following list is the possible topics for future research: (i) The algorithm for transforming the generalized stable reachability matrix into skeleton matrix has been proposed in chapter 3. However, be fore that we need to raise the power of the generalized one-step reachability matrix, whic h requires a large amount of computation. When the state set of the machine is large, this issue is more significant. If we can obtain a likely onestep skeleton matrix from the one-step reachability matrix and raise the power of this numerical skeleton matrix instead, then the calculation is mu ch simpler. But we need spend time to keep all the information that we need in the transforming and computation. (ii) The introduction of generali zed state transforms an async hronous machine with critical races into a deterministic machine. However, the state space is enlarged depending on the number of critical races. If we can minimize th e state space, then it will increase the speed of computation significantly too.

PAGE 75

75 (iii) Although we can construct an output fee dback controller for the closed loop system to eliminate the effects of critical race whenever a controller exists this controller may not be minimal. We can also work on this issue to fi nd out a good strategy to minimize the controller. (iv) The present discussion excludes the exis tence of infinite cycles in the existing machine. We shall deal with the situation when both critical races a nd infinite cycles occur in the defective machine. The controller constructed in the present work ensures that the closed-loop system in Figure 1-1 and Figure 2-3 operates in fundamental mode. The input changes were only allowed during stable combinations. This requires the re striction of the contro lled machine to those without any unstable cycles; otherwise the contro ller cant do anything to correct the machine once the machine enters a cycle.

PAGE 76

76 LIST OF REFERENCES Alpan, G. and Jafari, M.A., Synthesis of a closed-loop combin ed plant and controller model, IEEE Trans. on Systems, Man and Cybernetics vol. 32, no. 2, 2002, pp. 163-175. Barrett, G., and Lafortune, S., Bisimulation, the supervisory control problem, and strong model matching for finite state machines, Journal of Discrete Event Dynamic Systems Volume 8, number 4, 1998, pages 377-429. Chu, T., Synthesis of hazard-free co ntrol circuits from asynchronous finite state machines specifications, Journal of VLSI Signal Processing vol.7, no.1-2, 1994, p. 61-84. Datta, P.K., Bandyopadhyay, S.K. and Choudhury, A.K., "A graph theoretic approach for state assignment of asynchronous sequential machines International Journal of Electronics vol.65, no.6, 1988, p. 1067-75. Davis, A., Coates, B. and Stevens, K., The post office experien ce: designing a large asynchronous chip, Proceeding of the Twenty-Sixth Hawaii International Conference on System Sciences vol.1, pp. 409-418. Dibenedetto, M.D., Nonlinear strong model matching, IEEE Trans. Automatic Control vol. 39, 1994, pp. 1351-1355. Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., Model matching for finite state machines, Proceedings of the IEEE C onf. on Decision and Control vol. 3, 1994, pp. 3117-3124. Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., Strong model matching for finite state machines with nondeterministic reference model, Proceedings of the IEEE Conf. on Decision and Control vol. 1, 1995, pp. 422-426. Dibenedetto, M.D., Saldanha, A., and Sangiovanni-Vincentelli, A., Strong model matching for finite state machines Proceedings of European Control Conference vol. 3, 1995, pp. 2027-2034. Dibenedetto, M.D., Sangiovanni-Vincentelli, A., and Villa, T., Model matching for finite-state machines, IEEE Transactions on Automatic Control vol. 46, no. 11, 2001, pp. 17261743. Eilenberg, S., Automata, Languages, and Machines, Academic Press, NY, 1994. Furber, S.B., Breaking step the return of asynchronous logic, IEE Review 1993, pp. 159-162. Fisher, P.D., and Wu S. F., Race-free state assignments for synthesizing large-scale asynchronous sequential logic circuits, IEEE Transactions on Computers vol. 42, no. 9, 1993, pp. 1025-1034.

PAGE 77

77 Geng, X., Model matching for asynchronous sequential machines, Ph.D. Dissertation, Department of Electrical and Computer Engin eering, University of Fl orida, Gainesville, FL 32611, USA, 2003 Geng, X., Hammer, J., Input/output control of async hronous sequential machines, IEEE Trans. on Automatic Control Vol. 50, No. 12, pp 1956-1970. Hammer, J., On some control problems in molecular biology, Proceedings of the IEEE Conference on Decision and Control Vol. 4, 1994, pp. 4098-4103. Hammer, J., On the modeling and control of biological signaling chains, Proceedings of the IEEE Conference on Decision and Control Vol. 4, 1995, pp. 3747-3752. Hammer, J., On corrective control of sequential machines, International Journal of Control Vol. 65, No. 65, 1996, pp. 249-276. Hammer, J., On the control of incompletely described sequential machines, International Journal of Control Vol. 63, No. 6, 1996, pp. 1005-1028. Hauck, S., Asynchronous design methodologies: an overview, Proceedings of the IEEE vol. 83, no. 1, 1995, pp. 69-93. Higham, L. and Schenk, E., The parallel asynchronous recursion model, Proceedings of the IEEE Symposium on Parallel and Distributed Processing 1992, 310. Holcombe, W.M.L, Algebraic Automata Theory, Cambridge University Press New York, 1982. Hubbard, P. and Caines, P.E., Dynamical consistency in hierarchical supervisory control, IEEE Trans. on Automatic Control vol. 47, no. 1, 2002, pp. 37-52. Huffman, D.A., [1954a] The synthesis of sequen tial switching circuits, J. Franklin Inst. vol. 257, pp. 161-190. Huffman, D.A., The synthesis of sequen tial switching circuits, J. Franklin Inst ., vol. 257, 1954, pp. 275-303. Huffman, D.A., The design and use of hazard-free switching networks, Journal of the Association of Computing Machinery vol. 4, no. 1, 1957, pp. 47-62. Isidori, A., The matching of a prescribed linear input -output behavior in a nonlinear system, IEEE Trans. Automatic Control vol. AC-30, 1985, pp. 258-265. Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., Synthesis of FSMs : Functional Optimization, Boston, MA: Kluwer, 1997.

PAGE 78

78 Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., Implicit computation of compatible sets for state minimization of ISFSMs, IEEE Transactions on Computer Aided Design vol. 16, 1997, pp. 657-676. Kam, T., Villa, T., Brayton, R., and Sangiovanni-Vincentelli, A., Theory and algorithms for state minimization of nondeterministic FSMs, IEEE Transactions on Computer Aided Design vol. 16, 1997, pp. 1311-1322. Kohavi, Z., Switching and Finite Automata Theory, McGraw-Hill Book Company, New York, 1970. Koutsoukos, X.D., Antsaklis, P.J., Stiver, J.A. and Lemmon, M.D., Supervisory control of hybrid systems, Proceedings of the IEEE vol. 88, no. 7, 2000, pp. 1026-1049. Lavagno, L., Keutzer, K., and Sangicivanni-Vincentelli, A., Algorithms for synthesis of hazardfree asynchronous circuits, Proceedings of the 28th ACM/IEEE Conference on Design Automation 1991, pp. 302. Lavagno, L., Moon, C. W., and Sangiovanni-Vincentelli, A., Efficient heuristic procedure for solving the state assignment proble m for event-based specifications, IEEE Transactions on Computer-Aided Design of In tegrated Circuits and Systems Vol. 14, 1994, pp 45-60. Lin, B. and Devadas, S., Synthesis of hazard-free multilevel logic under multiple-input changes from binary decision diagrams, IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems vol. 14, no. 8, 1995, pp. 974-985. Lin, F., Robust and adaptive supervisory cont rol of discrete event systems, IEEE Transactions on Automatic Control vol. 38, n0. 12, 1993, pp. 1848-1852. Maki, G., and Tracey, J., A state assignment procedure for asynchronous sequential circuits, IEEE Transactions on Computers vol. 20, 1971, pp. 666-668. Marshall, A., Coates, B. and Siegel, F., Designing an asynchronous communications chip, IEEE Design & Test of Computers Vol. 11, no. 2, 1994, pp. 8-21. Mealy, G.H., A method for synthesizi ng sequential circuits, Bell System Tech. J ., vol. 34, 1955, pp. 1045-1079. Moon, C.W., Stephan, P.R., and Brayton, R.K., Synthesis of hazard-free asynchronous circuits from graphical specifications, IEEE International Confer ence on Computer-Aided Design 1991, pp. 322 Moore, B. and Silverman, L., Model matching by state feedback and dynamic compensation, IEEE Trans. Automatic Control vol. 17, 1972, pp. 491-497. Moore, E.F., Gedanken-experiments on sequential machines, Automata Studies, Annals of Mathematical Studies no. 34, Princeton University Press, N.J., 1956.

PAGE 79

79 Moore, S.W., Taylor, G.S., Cunningham, P.A., Mullins, R.D. and Robinson, P., Self-calibrating clocks for globally asynchronous locally synchronous systems, Proceedings of Inter. Confer. on Computer Design 2000, pp. 73-78. Morse, A.S., Structure and design of linear model following systems IEEE Trans. Automatic Control vol. 18, 1973, pp. 346-354. Murphy T.E., Geng X., and Hammer J., Controlling races in asynchronous sequential machines, Proceeding of the IFAC World Congress Barcelona, July 2002.. Murphy T.E., Geng X., and Hammer J., On the control of asynchronous machines with races, IEEE Transactions on Automatic Control vol. 48, no. 6, pp. 1073-1081. Murphy, T.E., On the control of asynchronous sequential machines with races, Ph.D. Dissertation, Department of El ectrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA, 1996. Nishimura, N., Efficient asynchronous simulation of a class of synchronous parallel algorithms, Journal of Computer and System Sciences vol. 50, no. 1, 1995, pp. 98-113. Nowick, S.M., Automatic synthesis of burst -mode asynchronous controllers, Ph.D. Dissertation, Stanford University, 1993. Nowick, S.M., and Coates, B., UCLOCK: automated design of high-performance unclocked state machines, ICCD '94. Proceedings., IEEE Inter national Conference on Computer Design 1994, pp. 434. Nowick, S.M., Dean, M.E., Dill, D.L. and Horowitz, M., The design of a high-performance cache controller: a case study in asynchronous synthesis, Proceedings of the 26th Hawaii International Conference on System Sciences 1993, pp. 419. Nowick, S.M. and Dill, D.L., Synthesis of asynchronous state machines using a local clock, IEEE International Conference on Computer Design 1991, pp. 192-197. Oliveira, D.L., Strum, M., Wang, J.C. and Cunha, W.C., Synthesis of high performance extended burst mode asynchronous state machines, Proceedings. 13th Symposium on Integrated Circuits and Systems Design 2000, pp. 41-46 Ozveren, C.M., Willsky, A.S., and Antsaklis, P.J., Stability and stabilizability of discrete event systems, J. ACM Vol. 38, 1991, pp. 730-752. Park, S.-J. and Lim, J.-T., Robust and nonblocking supervisor y control of nondeterministic discrete event systems using trajectory models IEEE Trans. on Automatic Control vol. 47, no. 4, 2002, pp. 655-658. Peterson, J.L., Petri Net Theory and The Modeling of Systems, Prentice-Hall, NJ, 1981.

PAGE 80

80 Ramadge, P.J.G., and Wonham, W.M., supervisory control of a class of discrete event processes, SIAM Journal of Control and Optimization vol. 25, no. 1, 1987, pp. 206. Ramadge, P.J.G., and Wonham, W.M., The control of discrete event systems, Proceedings of IEEE vol. 77, no. 1, 1989, pp. 81-98. Cole, R. and Zajicek, O., The expected advantage of asynchrony, Journal of Computer and System Science vol. 51, no. 2, pp. 286-300. Shields, M.W., An Introduction to Automata Theory, Blackwell Scientific Publications, Boston, 1987. Thistle, J. G. and Wonham, W.M., Control of infinite behavior of finite automata, SIAM Journal on Control and Optimization vol. 32, no. 4, 1994, pp 1075-1097. Unger, S. H., Asynchronous Sequential Switching Circuits, Wiley-Interscience, New York, NY, 1969. Unger, S. H., Self-synchronizing circuits and non-fundamental mode operation, IEEE Trans. Computers vol. 26, no. 3, 1977, pp. 278-281. Unger, S. H., Hazards, critical races, and metastability, IEEE Trans. on Computers vol. 44, no. 6, 1995, pp 754-768. Venkatraman, N., On the control of asynchronous sequent ial machines with infinite cycles, Ph.D. Dissertation, Department of Electrical and Computer Engineer ing, University of Florida, Gainesville, FL 32611, USA, 2004 Venkatraman, N., On the control of asynchronous sequent ial machines with infinite cycles, International Journal of Control Vol. 79, No. 07, 2006, pp. 764-785. Yu, M.L. and Subrahmanyam, P.A., A path-oriented approach for reducing hazards in asynchronous designs, Proceedings of Design Automation Conference 29th ACM/IEEE 1992, pp. 239

PAGE 81

81 BIOGRAPHICAL SKETCH Jun Peng was born in Wuhan, Hubei Province, China. She received her bachelors degree in automatic control and masters degree in control theory and cont rol engineering from Shanghai Jiao Tong University, Shanghai, China, in July 2000 and in April 2003, respectively. She began her Ph.D. program in the Department of Electrical and Computer Engineering at University of Florida, Gainesville, FL in August 2003. Her resear ch interests include asynchronous sequential circuits, application of asynchronous seque ntial systems in computer architecture, artificial intelligence and biological systems, control theory, control systems, and applications of control theory in computer co mmunication networks. She received her Ph.D. in August 2007.